Is there a decent explanation of what the conflict is about, that is understandable with a non-physics-major college level maths background?
Trying to give a slightly more high level idea, the conflict is simply that if you introduce the gravitational effects (spacetime bending) from general relativity into the equations of quantum mechanics you get infinities all over the place. This happens because in QM particles are not localized, they exist as a probability wave with certain values at any space and time, but the effects of their mass at all those spaces and times adds up to infinity if you just apply the calculations from relativity, in a way that we haven't been able to solve.
This is all very roughly speaking - the exact reasons are quite technical and I don't understand them myself. Electromagnetism and other interactions had similar issues, but we found mathematical solutions to fix them, and those solutions we know just don't work with the equations for gravitational interactions.
Most things interact with each other, they create "particles" to exchange force / energy (actually in the case here much of this is actually virtual particle interaction / creation). An example of this is when two magnets are interacting with each other, the repulsion/attraction that the magnets / and magnetic fields, the force carriers there actually described by best by "virtual photons" https://en.wikipedia.org/wiki/Virtual_photon#:~:text=Virtual....
The whole standard model is like this (protons/neutrons exchange gluons via the strong nuclear force).
For gravity, there isn't a good "particle" we've found that can accurately describe how gravity behave. It's good to note here a "particle" is what is used to "quantize" something, and so quantum mechanics is the study of how these subatomic particles interact and are created when things like protons are smashed together in a particle accelerator.
Gravity, on the other hand, behaves more like a classical theory (in the sense that it is a field, rather than a discrete quantized energy exchange). In the parlance of the field, mass, appears to cause a curvature in space time according to the rules (quite accurately predicted) in general relativity.
This appears to be some sort of more reconciliation by acknowledgment of the different aspects of the behaviors of the models (GR vs QM), and chooses a path to verify this by looking at the breakdown of matter (in this case, entangled quantum particles) in a asomtomic limit that might be imposed on a quantized theory when that is subatomically reconciled against a continuous field like gravity.
(My own notes) This might be indicating that there is a continuous nature (or at least continuous at the scales that these subatomic particles experience) such that when the particles interact with the continuous field, their movement creates some snapback of the field (thinking how movement in water creates microscopic cavitation) which disrupts what would be a continuous laminar flow in the water case, creating macroscopic turbulence (in this case disturbing the entangled particles)
* Edit: a little clarity / notes on the SM/QM/GR overlap
Like electron tunneling that made a good amplifier (transistor)?
Could this “snapback” be utilized in interesting ways?
If you want to go much deeper, Roger Penrose wrote The Road to Reality which is a much much harder read (I was never able to complete it after 3 separate attempts).
It doesn't go into math but mentions that in the standard model, interactions are local, which for example means something like photon emission is instantaneous. When applied to gravity/gravitons, you get infinities in the math. This sort of makes sense to me, but I have yet to find anything that does an understandable job explaining how/why exactly the math breaks down.
There's a series on the maths of general relativity [2] which I haven't completed but I'm hoping will provide a bit more background to understand the math.
Also, the article mentions that the new theory was motivated by solving the black hole information paradox [3], which there's a good video [4] about as well
[1] https://www.youtube.com/watch?v=n7cOlBxtKSo
[2] https://www.youtube.com/watch?v=xodtfM1r9FA&list=PLu7cY2CPiR...
[3] https://en.wikipedia.org/wiki/Black_hole_information_paradox