There are several results by James Arthur that fall into this category.  (I didn't mention them previously because I thought they were already mentioned by Kevin Buzzard in the talk cited by the OP, but I belatedly realized that I was confusing that with <a href="http://www.andrew.cmu.edu/user/avigad/meetings/fomm2020/slides/fomm_buzzard.pdf">a different talk by Buzzard</a>.)

On page 13 of <a href="https://arxiv.org/abs/1812.09269">Abelian Surfaces over totally real fields are Potentially Modular</a> by George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, there is the following remark.

> It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of GSp<sub>4</sub>, as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT18], but this proof is only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.

Arthur's (unavailable) references [A24] through [A27] are:

[A24] <I>Endoscopy and singular invariant distributions</I>, in preparation.

[A25] <I>Duality, Endoscopy, and Hecke operators</I>, in preparation.

[A26] <I>A nontempered intertwining relation for $GL(N)$</I>, in preparation.

[A27] <I>Transfer factors and Whittaker models</I>, in preparation.