Conditional Non-Soficity of p-adic Deligne Extensions: on a Theorem of Gohla and Thom

A long standing problem asks whether every group is sofic, i.e., can be separated by almost-homomorphisms to the symmetric group Sym(n). Similar problems have been asked with respect to almost-homomorphisms to the unitary group U(n), equipped with various norms. One of these problems has been solved for the first time in [De Chiffre, Gelbsky, Lubotzky, Thom, 2020]: some central extensions Γ˜ of arithmetic lattices Γ of Sp(2g,Qp) were shown to be non-Frobenius approximated by almost homomorphisms to U(n). Right after, it was shown that similar results hold with respect to the p-Schatten norms in [Lubotzky, Oppenheim, 2020]. It is natural, and has already been suggested in [Chapman, Lubotzky, 2024] and [Gohla, Thom, 2024], to check whether the Γ˜ are also non-sofic. In order to show that they are (also) non-sofic, it suffices:
(a) To prove that the permutation Cheeger constant of the simplicial complex underlying Γ is positive, generalizing [Evra, Kaufman, 2016]. This would imply that Γ is stable.
(b) To prove that the (flexible) stability of Γ implies the non-soficity of Γ˜.
Clause (b) was proved by Gohla and Thom. Here we offer a more algebraic/combinatorial treatment to their theorem.