Brady–McCammond's CAT(0) conjecture for the orthoscheme complex of non-crossing partitions

Let $\operatorname{NC}(n)$ be the lattice of non-crossing partitions, and let its orthoscheme complex be its simplicial realisation equipped with the orthoscheme metric obtained by identifying each maximal simplex with the $n$-orthoscheme. Let $\mathcal{B}_n$ denote the braid group on $n$ strands. Brady–McCammond's conjecture. For every $n$, the orthoscheme complex of $\operatorname{NC}(n)$ is $\operatorname{CAT}(0)$, and consequently the braid group $\mathcal{B}_n$ is $\operatorname{CAT}(0)$. This geometric formulation is motivated by the product decomposition of the relevant universal cover and the fact that $\operatorname{CAT}(0)$ curvature of the orthoscheme complex would imply the required $\operatorname{CAT}(1)$ link condition. The supplied text does not state whether the conjecture has been resolved.