Product formula conjecture for Dunkl–Bessel functions

Let $N\geq 1$, let $\mathcal{R}\subset\mathbb{R}^N$ be a finite root system, and let $\theta\in\theta(\mathcal{R})$ be nonnegative. For $a_1,a_2\in\mathbb{R}^N$, consider the Dunkl–Bessel functions $J^{\mathcal{R}(\theta)}_{a_1}$ and $J^{\mathcal{R}(\theta)}_{a_2}$. Product formula conjecture. There exists a nonnegative probability measure $\mu^{\mathcal{R}(\theta)}_{a_1,a_2}$ on $\mathbb{R}^N$ such that

$$J^{\mathcal{R}(\theta)}_{a_1}(x)J^{\mathcal{R}(\theta)}_{a_2}(x)=\int_{\mathbb{R}^N}J^{\mathcal{R}(\theta)}_a(x)\\,d\mu^{\mathcal{R}(\theta)}_{a_1,a_2}(a)$$

for all $x\in\mathbb{C}^N$. This is a positivity conjecture for the product formula of Bessel functions associated with finite root systems; the source describes it as well known and uses it as an assumption for subsequent weak-convergence results.