Module-isomorphism conjecture for regular posets

Let $\sfRP_n$ be the class of regular posets on $[n]$, let $\sfM_P$ be the associated $0$-Hecke module, and let $\Sigma_L(P)$ be the set of linear extensions of $P$. Write $\Deq$ for descent-preserving isomorphism of the relevant subsets of the symmetric group.

Module-isomorphism conjecture. For $P,Q\in\sfRP_n$, if

$$\sfM_P\cong\sfM_Q,$$

then

$$\Sigma_L(P)\Deq\Sigma_L(Q).$$

The conjecture seeks to extend the classification of modules from regular Schur labeled skew shape posets to all regular posets. The supplied text presents it as unresolved and provides supporting evidence from special cases.