Scott–Sokal conjecture for elementary symmetric polynomials

For integers $m,n$ with $2\leq m\leq n$, let $E_{m,n}(x)=\sum_{1\leq i_1<\dots<i_m\leq n}x_{i_1}\cdots x_{i_m}$ be the $m$th elementary symmetric polynomial, and consider its negative power on the positive orthant $\mathbb{R}_{>0}^n$. Scott–Sokal's conjecture. The function $E_{m,n}^{-\alpha}$ is completely monotone on $\mathbb{R}_{>0}^n$ if and only if $\alpha=0$ or $\alpha\geq (n-m)/2$. Scott and Sokal gave this conjectural parameter description; the paper proves the preceding Riesz-kernel conjecture for all elementary symmetric polynomials, but the source does not state here whether this exact conjecture is resolved.