The centralizer braid-group conjecture for regular elements

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, put $w=\beta({\bf w})$, and let $C_W(wF)$ be the centralizer of the corresponding regular element of $WF$. Let $B(w)$ be the braid group of the complex reflection group $C_W(wF)$, and let $\gamma:B(w)\to C_B({\bf w}F)$ be the natural homomorphism whose image under $\beta$ is $C_W(wF)$. Centralizer braid-group conjecture. The morphism $\gamma$ is an isomorphism. The paper notes that this is known in several cases, including split types $A$ and $B$, Coxeter elements in split groups, and fourth roots in type $D_4$, but leaves the general assertion open.