The Riesz-kernel conjecture for complete hyperbolic polynomials

Let $p\in \mathbb{R}[x_1,\dots,x_n]$ be a complete hyperbolic polynomial with hyperbolicity cone $C$. For a real $\alpha>n$, let $q_{\alpha}$ be the function defined by

$$q_{\alpha}(y)=(2\pi)^{-n}\int_{\mathbb{R}^n}p(e+i\cdot x)^{-\alpha}e^{\langle y,e+i\cdot x\rangle}\,dx,$$

where $e\in C$, and let $C^*$ denote the dual cone. The Riesz-kernel conjecture. There exists a real $\alpha>n$ such that $q_{\alpha}(y)$ is nonnegative on $C^*$. In particular, $q_{\alpha}$ is a Riesz kernel. The conjecture is known for hyperbolic polynomials admitting a symmetric determinantal representation, and the paper proves it for all elementary symmetric polynomials; the general case remains open.