Bessis–Digne–Michel's centralizer conjecture for regular braid-group elements

Let $V$ be a finite-dimensional complex vector space, let $W\subset\operatorname{GL}(V)$ be an irreducible finite complex reflection group, and let ${\mathcal{M}}$ be the complement of its reflecting hyperplanes. Let $\zeta\in\mathbf{C}$ be a root of unity of order $d$, where $d$ is regular for $W$, and choose $w\in W$ such that

$$\operatorname{ker}(w-\zeta\operatorname{Id})\cap{\mathcal{M}}\neq\emptyset.$$

Write ${\mathcal{M}}(w)=\operatorname{ker}(w-\zeta\operatorname{Id})\cap{\mathcal{M}}$, let $C_W(w)$ be the centralizer of $w$ in $W$, and let $\mathbf{B}$ and $\mathbf{B}_{C_W(w)}$ be the braid groups associated with $W$ and $C_W(w)$, respectively. Choose a basepoint in the image of ${\mathcal{M}}(w)$, and let $\mathbf{w}\in\mathbf{B}$ be the corresponding lift of $w$.

Bessis–Digne–Michel's centralizer conjecture. The natural morphism

$$\mathbf{B}_{C_W(w)}\longrightarrow\mathbf{B}$$

induces an isomorphism between $\mathbf{B}_{C_W(w)}$ and the centralizer $C_{\mathbf{B}}(\mathbf{w})$.

This conjecture asserts that the Springer-theoretic centralizer of a regular element lifts from the reflection group to its braid group. The source states that it is checked for Artin braid groups and some other cases; no general resolution is given here.