Positivity conjecture for the multivariate Bessel addition law

Let $\theta>0$, and let $\a=(a_1\le\dots\le a_N)$ and $\b=(b_1\le\dots\le b_N)$ be $N$-tuples. The distribution $\c=\a+_\theta\b$ is defined through the multivariate Bessel identity

$$\mathbb E B_{(c_1,\dots,c_N)}(x_1,\dots,x_N;\,\theta)=B_{(a_1,\dots,a_N)}(x_1,\dots,x_N;\,\theta)B_{(b_1,\dots,b_N)}(x_1,\dots,x_N;\,\theta).$$

Positivity conjecture. For every such $\a$ and $\b$, there exists a probability measure on $N$-tuples $\c=(c_1\le\dots\le c_N)$ satisfying the identity for all $x_1,\dots,x_N\in\mathbb C$.

The identity is known to define a compactly supported generalized function or distribution of total mass $1$, but it is not known in general whether this distribution is positive and hence a probability measure. The conjecture would provide a probabilistic interpretation of the Bessel addition law and is relevant to addition problems for general beta random matrices.