The optimality conjecture for the hyperbolic-polynomial approximation factor

Let $PHYP(n,m)$ be the set of positive hyperbolic polynomials of degree $m$ in $n$ variables, let $S_{n,n-m}$ be the polynomial used in the source, and let $\operatorname{Cap}$ denote capacity. Define

$$\gamma(n,m)=\inf_{r\in UPHYP(n,n-m)}\sup_{q\in PHYP(n,m)}\frac{\operatorname{Cap}(qr)}{\frac{\partial^n}{\partial x_1\cdots\partial x_n}q(0,\ldots,0)r(0,\ldots,0)}.$$

The optimality conjecture. The particular polynomial $S_{n,n-m}$ is optimal in the sense that

$$\gamma(n,m)=\sup_{q\in PHYP(n,m)}\frac{\operatorname{Cap}(qS_{n,n-m})}{\frac{\partial^n}{\partial x_1\cdots\partial x_n}q(0)S_{n,n-m}(0)}=\frac{n^m}{m!}.$$

The preceding discussion establishes the lower bound $\gamma(n,m)\geq n^m/m!$; the displayed equality would identify the exact approximation factor for the chosen class of positive hyperbolic polynomials. The source does not provide a resolution.