The CAT(0) conjecture for Coxeter simplex splittings

Let $W$ be a Coxeter group, let $S'\subset S$ be an infinite special subgroup of rank $v+1$, and let $\Lambda$ be the $v$-splitting of $W$ determined by $S'$, obtained by adding the missing generators to the simplex-of-groups construction. The universal cover of this splitting is the simply connected complex on which $W$ acts with quotient a $v$-simplex.

CAT(0) conjecture. The universal cover of the $v$-splitting of $W$ determined by $S'$ is $\operatorname{CAT}(0)$.

This conjecture would supply the nonpositive-curvature input needed to analyze the FAn conjecture for Coxeter groups. The paper proves the corresponding assertion in dimension $2$, but the general statement remains open.