The symmetrization conjecture for Rayleigh partition functions

Let $E$ be a finite set of size $m$, let $\omega$ be a weight function, and let $Z(\omega;\mathbf y)$ be its partition function. Define the symmetrized weight and partition function by

$$\widetilde Z(\omega;\mathbf y)=\frac1{m!}\sum_{\sigma\in\mathfrak S_m}Z(\omega;\mathbf y_\sigma).$$

Symmetrization conjecture. If $Z(\omega;\mathbf y)$ is Rayleigh, then $\widetilde Z(\omega;\mathbf y)$ is Rayleigh. This is stated as an equivalent form of the Big Conjecture and remains open.