Dyer's finite Coxeter group conjecture on joins in weak order

Let $(W,S)$ be a finite Coxeter system, let $T$ be its set of reflections, and let $\ell$ be the Coxeter length function. For $w\in W$, define its left-reflection set by

$$T_L(w)=\\{t\in T\mid \ell(tw)<\ell(w)\\}.$$

For $u,v\in W$, let $V_W(u,v)$ be the set of vertices of directed Bruhat paths starting at the identity whose edge labels lie in $T_L(u)\cup T_L(v)$. Dyer's finite join conjecture. For all $u,v\in W$,

$$T_L(u\vee_R v)=T\cap V_W(u,v).$$

Here $u\vee_R v$ denotes the join in the right weak order. This reformulates Dyer's conjectural description of joins in the extended weak order in terms of left-reflection sets and Bruhat paths; the source states that the original conjecture remains open even for finite Coxeter systems.