Dyer's conjecture on joins of biclosed sets in extended weak order

Let $(W,S)$ be a Coxeter system with associated root system $\Phi$, positive roots $\Phi^+$, and reflection $s_\alpha$ corresponding to each $\alpha\in\Phi^+$. Let $\mathscr{B}(\Phi^+)$ be the poset of biclosed subsets of $\Phi^+$ ordered by inclusion. For $A\subseteq\Phi^+$, define $\tau(A)$ to be the set of elements $w\in W$ admitting a reduced expression whose successive simple-reflection products strictly increase in length and whose associated positive roots all lie in $A$. Dyer's conjecture. For $A,B\in\mathscr{B}(\Phi^+)$, their join in $(\mathscr{B}(\Phi^+),\subseteq)$ is

$$J(A,B)=\\{\alpha\in\Phi^+\mid s_\alpha\in\tau(A\cup B)\\}.$$

The conjecture gives an explicit characterization of joins in the extended weak order. The surrounding text states that the lattice conjecture for this poset is known for affine types, but that this join characterization remains open even for finite Coxeter systems.