Conjecture on the potential kernel for the radial Dunkl Laplacian

Let $B^+$ and $S^+$ denote the positive chambers in the unit ball and on its boundary, respectively. Write $P^W(x,y)$ for the Poisson kernel, let $\Phi_+$ be the set of positive roots, let $\sigma_\alpha$ be the reflection associated with $\alpha\in\Phi_+$, and let $k$ be the multiplicity function. Write $f\asymp g$ when there are constants $0<C_1\leq C_2$ depending only on the dimension, the root system, and $k$, such that $C_1f\leq g\leq C_2f$ on the common domain of $f$ and $g$. Define

$$\Omega(x,y)=\frac{1-|x|^2}{|x-y|^d\prod_{\alpha\in\Phi_+}|x-\sigma_\alpha y|^{2k(\alpha)}}.$$

Potential-kernel conjecture. For $x\in B^+$ and $y\in S^+$,

$$P^W(x,y)\asymp\Omega(x,y).$$

This conjecture gives the expected sharp comparability of the Poisson kernel for the radial Dunkl Laplacian in a positive chamber. The supplied text does not state whether it is open, solved, or refuted.