Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$ superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $\mathcal X$. I don't really know what that would entail; apparently we have partial information about this theory, suggesting that it's difficult to describe even in a physics sense (e.g. non-Lagrangian), let alone to mathematically formalize.
But I have also heard from mathematicians who are interested in Theory $\mathcal X$ for what I assume are entirely mathematical reasons. I came away with the impression that even though we can't construct it, there are ways to study it to yield interesting results in pure mathematics; but I don't know any examples of such results.
So my question is: what are some purely mathematical takeaways from the story of Theory $\mathcal X$? And, if it's known, what would be some expected mathematical consequences of a construction of Theory $\mathcal X$ in a physics sense?
I get the impression that various dimensional reductions of Theory $\mathcal X$ should include several commonly-studied TQFTs and QFTs, so in a sense studying Theory $\mathcal X$ generalizes the study of those TQFTs. So one possible answer is that there could be theorems about those TQFTs whose proofs were inspired by some conjectured aspect of Theory $\mathcal X$ — but is this accurate?