Earlier quoted context omitted.
As far as I know, there isn't much theory behind the "important work" being done here, just "let's smash stuff at higher energy and see what shakes out." Science is supposed to go the other way. You are supposed to have a theory or a set of theories and then use experiments to test them. If this experiment can falsify the standard model in a meaningful way, that's cool, but it's a lot better if it can actually prove…
I hope this doesn't seem abrupt, and when/if it does, that it comes across as coming from a place inviting further discussion and appreciating your curiosity. That's one of the effects of the pitch framing by Sabine: the design intent is not about smashing stuff at higher energy , despite the basis of the critique requiring that it is . My comment above is really long, but tl;dr the impetus for both facilities is sma…
The scientific goals of that experiment are somewhat more unclear, though. The LHC had a landmark scientific purpose, finding the last particle in the standard model. There is, as far as I can tell, no specific experiment that can be the headline for this new machine because the LHC pretty much did its job (modulo some error) and string theory et al need higher energy. There are a bunch of guesses about the higgs field and about dark matter that failed to materialize at the LHC, so now we want moar energy to see if that fixes our problems.
As to the theory they will be proving, maybe there are a few minor ones about the higgs field, but that's pretty much it at this point.