Barycentric subdivision conjecture for exterior algebraic shifting of surfaces

Let $K$ be a surface triangulation, let $\operatorname{sd}(K)$ denote its barycentric subdivision, and let $\Delta^{ex}$ denote exterior algebraic shifting over a fixed field. A homology lex-segment is a simplicial complex whose faces form the corresponding homology lex-segment.

Barycentric subdivision conjecture. If $K$ is a surface triangulation, then $\Delta^{ex}(\operatorname{sd}(K))$ is a homology lex-segment.

This is proposed as the surface analogue of the deterministic statement known for barycentric subdivisions of graphs. It would provide a possible route toward the conjectured concentration result for uniform triangulations of fixed surfaces; the source gives no resolution.