General semisimple-group conjecture for totally positive varieties

Let $G$ be a semisimple algebraic group, and let $Y_{[u,v]}$, $Y_u^\circ$, ${\rm Lk}(u,v)$, and the strata $S_{u,v,w}$ be the stratified spaces and links associated with Bruhat intervals $[u,v]$ in the paper. Theorems on the contraction of links and the product decomposition of $Y_{[u,v]}$ are understood in their stated generality for type $A$ and the relevant varieties.

General semisimple-group conjecture. Theorems on the contractions and product structure—and consequently the orientability theorem and the relative-homology corollary—hold for any semisimple algebraic group $G$.

The claim proposes that these topological properties of totally positive varieties extend from the type-$A$ setting to arbitrary semisimple algebraic groups. The source gives no evidence of a resolution, so the conjecture is recorded as open.