There's not much to verify there. They claim they got a way to use electromagnetic time crystals as an electromagnetic sensor. It's not an extraordinary claim, but they don't exactly say how either, so we can't check.

And if they improve their sensor into something that seems viable, it would be good at detecting the minuscule fields that some candidate dark matter particles should create very rarely when they interact with something. Maybe becoming a better detector for them than the current state of the art (this, I think: https://en.wikipedia.org/wiki/Axion_Dark_Matter_Experiment).

The keyword you want to look for is 'ultra light bosonic dark matter'. This search is legit.

This paper will send you almost in the right direction. The coupling is different, but the rest of the physics is the same: https://arxiv.org/abs/1512.06165

No idea about the veracity of it, but it sounds almost like it's a dark matter/gravitational analogue to the piezo electric effect. That's a very interesting idea, it'd create a signal from a dark matter partical (assuming they are particals) interacting gravitationally and disrupting the crystal temporarily.