The writing is a bit ambiguous, but if I'm parsing it correctly it's only a big deal to the small subset of physicists who care about mathematical rigor: proof that Quantum Field Theory (i.e. what theorists have been using since the 1940s or so) is mathematically sound.

This part,

The existence proof pertains to the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model

looks like a reference to SU(5) GUT from 1974:

https://en.wikipedia.org/wiki/Georgi%E2%80%93Glashow_model

As a slightly less but still very casual enjoyer too, I don't think there is a huge amount of value in dumping the lean proof or the natural language proof. I'd rather have an abstract of a paper that I can at least hope to understand 75% of before getting lost.

I'm guessing if you were really interested, you could email YANGMILLS4D@gmail.com (itself a hilarious e-mail address).

Careful, completion of the standard model ≠ theory of quantum gravity or theory of everything. The actual question is too abstruse for me (biophysicist).