The Skordis & Zlosnik paper can be found at https://arxiv.org/abs/2007.00082
I'll start with some comments for non-experts, and then escalate about ten paragraphs further down.
tl;dr: there are lots of things that hang on General Relativity being correct practically everywhere in our universe, the only "hiding places" that don't break astrophysical objects (or even laboratory experiments, including in robotic laboratories we have sent to other places in our solar system) is very close to the big bang, deep inside black holes, and in large masses brought into a quantum superposition of space.
The core of General Relativity is a mathematically-complete relationship between matter in a spacetime, and spacetime curvature. The key point is that the exact relationship can be described correctly at every point in the spacetime.
The relationship takes on a particular form: the Einstein Field Equations (EFE), which can be written (omitting prefactors, the cosmological constant, and indices) as G = T, where "G" is the Einstein curvature tensor and T is the stress-energy tensor. G and T are really tensor fields, which take on a value at every point in a spacetime. T encodes the matter content at a given point, and can represent incredibly complex "piles" of interacting particles or matter field values.
The slogan form of this is, "moving matter generates curvature, curvature moves matter". The result is that setting out distribution of matter that obeys some dynamical laws generates spacetime curvature. But there can additionally be "vacuum-generated" curvature around which one might sprinkle some matter, which would then be entrained into orbits or other trajectories by that curvature. The latter is the approach used in the study of theoretical black holes, for example. But also it's a good way to understand the expansion of space: there is a "vacuum-generated" expanding spacetime curvature, with galaxy clusters distributed through it, and entrained into trajectories principally by that curvature.
We can also look at trajectories, and with the assumption that all matter falls the same way ("the universality of free-fall"), recover the curvature, the distribution and dynamics of the stress-energy, or both. This is quite common in astrophysical settings, where one is trying to determine an equation of state for a body like a white dwarf or a neutron star.
One of the problems that particle Dark Matter seeks to solve is that gas and stars far from the centres of galaxies of do not fall in faster than they do. Something holds them up, keeping them on unexpected (non-Newtonian) trajectories. We can model this by introducing extra complexity into the stress-energy tensor, starting with something simple and abstract and drilling down where evidence allows us to do so. One might write this as G = f(T), some function on the dark-matter-free matter content matches the curvature exactly. We can improve upon an initial simple f() over time. But alternatively we can explore f(G) = T.
It's been known since the 1920s that the relationship at the core is pretty flexible: we can choose a background curvature (flat spacetime, a black hole) and add matter and see what happens, or we can start with a distribution of matter (and dynamics) and see what it does to the Einstein curvature tensor. We can (a) encode more and more complicated representations of matter into the stress-energy tensor, and keep curvature simple. Or instead, (b) we can adapt the curvature making the "background" more and more complicated, and then sprinkle a relatively simple distribution of matter on top.
(Particle) Dark Matter is mainly the (a) approach. We assume that a galaxy's curvature is fairly simple in the bulk, lay out a reasonable model of the bulk visible matter of a galaxy (and electromagnetic radiation, and neutrinos), and then ask, "What must we do to the stress-energy tensor so that we still generate the observed trajectories of outermost matter (gas, dust, stars)?".
MOND is mainly the (b) approach. There is an extremely simple empirical law (from Mordehai Milgrom in 1981) that adapts Newtonian gravitation to generate the non-Newtonian orbits of outermost stars observed in most spiral galaxies. This may translate into a function on the Einstein curvature tensor that produces a "background" in which a realistic description of a galaxy's visible matter (and electromagnetic radiation and neutrinos) is kept from being flung out into intergalactic space. The problem is that it turns out one cannot do this while keeping General Relativity's exact relationship between matter and curvature, because keeping Milgrom's constant means matter at the outsides of galaxies feels gravitation (and other interactions) differently from matter in for example the solar circle (the part of the Milky Way where we find our sun's orbit, about 8.5 kiloparsecs from the core), or in the central parsec.
I'll expand on this by quoting [1] (Stacey McGaugh, second author, is a MOND proponent quoted in the aps.org article), "The heart of GR is the equivalence principle(s), in its weak (WEP), Einstein (EEP) and strong (SEP) form. The WEP states the universality of free fall, while the EEP states that one recovers special relativity in the freely falling frame of the WEP. These equivalence principles are obtained by assuming that all known matter fields are universally and minimally coupled to one single metric tensor, the physical metric. It is perfectly fine to keep these principles in MOND, although certain versions can involve another type of (dark) matter not following the same geodesics as the known matter, and thus effectively violating the WEP. Additionally, note that the local Lorentz invariance of special relativity could be spontaneously violated in MOND theories. The SEP, on the other hand, states that all laws of physics, including gravitation itself, are fully independent of velocity and location in spacetime [...] This principle has to be broken in MOND."
SEP also means WEP holds, and WEP requires that gravitational mass and inertial mass are identical for all bodies including self-gravitating ones like planets, stars, neutron stars, and so on.
A few pages along [2], Famaey & McGaugh write, "It is perhaps more important that, if MOND is correct in the sense of the acceleration a_0 [Milgrom's law's constant] being a truly fundamental quantity, the strong equivalence principle cannot hold anymore, and local Lorentz invariance could perhaps be spontaneously violated too."
That is, relativistic MOND generally means you lose the guarantee of Special Relativity holding in a small neighbourhood around every point in spacetime, which is liable to affect tests of the Standard Model of Particle Physics (which has that guarantee fundamentally baked in). Worse than that, with SEP violation, in general laws of physics must vary depending on a probe's proximity to mass, particularly probed bodies' response to acceleration. This is awkward given recent astrophysical evidence supporting the SEP (e.g. [3]).
In order to accord with evidence in favour of the SEP, one has to do some handstands, adding complexity to the function on curvature.
Of note to experts is Skordis & Zlosnik p. 5: "The vector in (5) does not seem to obey gauge invariance but in the quadratic action (13) it does so through mixing with diffeomorphisms of h_munu" and "The resulting action is that of the gauged ghost condensate (GGC) [122] or bumblebee field [123, 124] which has been proposed as a healthy gauge-invariant theory of spontaneous Lorentz violation."
from which one can jump into https://en.wikipedia.org/wiki/Bumblebee_models#Nambu%E2%80%9... (which is decently encyclopedic) and relate that to the quote at the top's argument that in a relativistic MOND, either Special Relativity isn't a guarantee in the small neighbourhood around every point (it is guaranteed by General Relativity, and is highly tested) or we recover it by adding more fields to the replacement of the \Lambda-equipped Einstein-Hilbert action.
I gather the idea is to suppress significant violations of the Strong Equivalence Principle, and to do so by adding yet more degrees of freedom.
From Famaey & McGaugh again [at [2]], "it is true that it would be more elegant to avoid too many additional degrees of freedom", which we can relate to K Freese's quote in the aps.org article.
Finally, quoting Skordis & Zlosnik again, "Studies of MOND with galaxy clusters [...] report that either a_0 is larger in clusters and/or an additional dark component is necessary even when the MOND prescription is used [...] the theory presented here has additional features warranting its separate testing with clusters." [Emphasis mine]
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[1] Famaey & McGaugh https://arxiv.org/abs/1112.3960 §7 p. 88.
[2] ibid., §10 p. 122
[3] Scott Ransom's 2014 slides on PSR J0337+1715 https://websites.utdallas.edu/nsm/texas2013/proceedings/1/2/...
later detailed observations (Ransom, Stairs, Archibald et al 2014) https://doi.org/10.1038%2Fnature12917 == https://arxiv.org/abs/1401.0535
test of Strong Equivalence Principle (Archibald et al 2018) https://doi.org/10.1038%2Fs41586-018-0265-1 == https://arxiv.org/abs/1807.02059