Nonnegativity conjecture for Riesz kernels

Let $f$ be a homogeneous hyperbolic polynomial, let $K$ be its hyperbolicity cone, and let $q_\alpha(\sigma)$ be the Riesz kernel defined by the integral representation

$$f(\theta)^{-\alpha}=\int_K \exp(-\langle\theta,\sigma\rangle)q_\alpha(\sigma)\,\mathrm{d}\sigma.$$

Nonnegativity conjecture for Riesz kernels. For $\alpha\gg 0$, the Riesz kernel $q_\alpha(\sigma)$ takes nonnegative values on $K$.

If true, this would make $q_\alpha(\sigma)\,\mathrm{d}\sigma$ a measure on $K$ and allow the construction of hyperbolic exponential families. The supplied text notes that nonnegativity has been suggested by various authors but that no proof was known there; the condition $\alpha>d$ is sufficient for the kernel to be well-defined, but is not asserted to be necessary.