Dyer's complete ortholattice conjecture for biclosed sets

Let $(W,S)$ be a Coxeter system with positive root system $\Phi^{{\scriptscriptstyle +}}$, and let $\mathcal B$ be the poset of biclosed subsets of $\Phi^{{\scriptscriptstyle +}}$, ordered by inclusion. For $A\subseteq\mathcal B$, write $\overline{\bigcup A}$ for the $2$-closure of its union. Dyer's conjecture. The poset $(\mathcal B,\subseteq)$ is a complete ortholattice. The join of a family $A\subseteq\mathcal B$ is

$$\bigvee A=\overline{\bigcup A},$$

and the ortholattice complement is the set complement in $\Phi^{{\scriptscriptstyle +}}$. This extends the corresponding result for finite Coxeter groups to infinite Coxeter groups.