Local structure conjecture for Dunkl weight functions

Let $W$ be a finite real reflection group acting on $\mathfrak{h}$, let $c\in\mathfrak{p}_{\mathbb{R}}$ be real, and let $\lambda\in\operatorname{Irr}(W)$. For $b\in\mathfrak{h}_{\mathbb{R}}$, set $W'=\operatorname{Stab}_W(b)$, let $\mathfrak{h}_{\mathbb{R},W'}$ be the real reflection representation of $W'$, and write $\mathfrak{h}_{\mathbb{R}}=\mathfrak{h}_{\mathbb{R},W'}\oplus\mathfrak{h}_{\mathbb{R}}^{W'}$. Write $x=(x',x”)$ accordingly, and let $K_{c,\lambda}(x)$ be the Dunkl weight function. Local structure conjecture. There exists a $W'$-equivariant $\operatorname{End}_{\mathbb{C}}(\lambda)$-valued analytic function $B(x)$ defined near $b$, with $B(b)=\operatorname{Id}$, such that

$$B(x)^\dagger K_{c,\lambda}(x)B(x)=\sum_{\mu\in\operatorname{Irr}(W')}K_{c,\mu}(x')\otimes h_\mu,$$

where each $h_\mu$ is a Hermitian form on $\operatorname{Hom}_{W'}(\mu,\lambda)$. This generalizes the known regular-point description and is established in the source when $b$ is regular, when $b=0$, and at a generic point of a reflection hyperplane; the general local description remains open.