Dyer's Bruhat-preclosure conjecture for joins in finite Coxeter groups

Let $W$ be a finite Coxeter group, and let $u,v\in W$. For $w\in W$, write $T_L(w)$ for its left-reflection set. Given $A\subseteq T$, define its Bruhat preclosure by

$$\overline{A}=\{t\in T\mid\text{there is an $A$-Bruhat path from }e\text{ to }t\}.$$

Here an $A$-Bruhat path is a directed path in the Bruhat graph from the identity element $e$, with every edge labeled by an element of $A$. Dyer's Bruhat-preclosure conjecture. For all $u,v\in W$,

$$T_L(u\vee_R v)=\overline{T_L(u)\cup T_L(v)}.$$

The conjecture gives a description of joins in the right weak order in terms of Bruhat preclosure. It is not known whether it holds for all finite Coxeter systems; the source reports proofs in types $I$ and $A$ and verification in types $F_4$, $H_3$, and $H_4$, while the type $B$ case motivates the paper.