Gurvits's inner-product conjecture for homogeneous real stable polynomials

Let $p,q \in \mathbb{R}_+[x_1,\ldots,x_n]$ be homogeneous real stable polynomials of total degree $d$. For a multi-index $\mu$ with $|\mu|=d$, write $p_\mu$ and $q_\mu$ for the corresponding coefficients, let $\binom{d}{\mu}$ denote the multinomial coefficient, and let $\alpha^\alpha=\prod_{i=1}^n\alpha_i^{\alpha_i}$. For a polynomial $f$, define its capacity by $\operatorname{Cap}_\alpha(f)=\inf_{x_i>0}f(x_1,\ldots,x_n)/(x_1^{\alpha_1}\cdots x_n^{\alpha_n})$. Gurvits's conjecture.

$$\sum_{|\mu|=d}\binom{d}{\mu}^{-1}p_\mu q_\mu\geq\frac{\alpha^\alpha}{d^d}\operatorname{Cap}_\alpha(p)\operatorname{Cap}_\alpha(q).$$

This conjecture seeks an inner-product lower bound for homogeneous real stable polynomials in terms of their capacities, extending capacity inequalities connected with stable polynomials and strong Rayleigh measures. The source does not state a resolution, so its status remains open.