The join expression conjecture for intersections of Verma modules

Let $(W,S)$ be a Weyl group with Bruhat order. For $w\in W$, let $\mathbf{JM}(w)$ be the distinguished set of Bruhat-maximal join-irreducible elements whose join is $w$, and let $\Delta_w$ denote the corresponding Verma submodule of the antidominant Verma module $\Delta_e$. Join expression conjecture. For each $w\in W$, we have

$$\Delta_w=\bigcap_{x\in\mathbf{JM}(w)}\Delta_x,$$

where the intersection is taken inside the ungraded module $\Delta_e$. The statement is known in type $A$, while examples in type $E_6$ show that the analogous assertion for arbitrary join expressions can fail; the conjecture concerns the distinguished expression $\mathbf{JM}(w)$ and remains open in general.