Uniform surface conjecture for exterior algebraic shifting

Let $S_g$ be a fixed orientable surface of genus $g$, and let $U_n(S_g)$ denote its uniform random triangulation with $n$ vertices. Let $\Delta^{ex}$ denote exterior algebraic shifting over a fixed field, and let $\Delta(S_g,n)$ be the corresponding homology lex-segment complex.

Uniform surface conjecture. For every fixed genus $g$, a.a.s.,

$$\Delta^{ex}(U_n(S_g))=\Delta(S_g,n).$$

This would show that the exterior shifting of a uniform triangulation of a fixed orientable surface is a homology lex-segment and, in particular, that $U_n(S_g)$ is a.a.s. $K_6$-free. The statement is not known; local neighborhoods of almost every vertex are known to be a.a.s. planar.