The averaged symmetrization inequality for Rayleigh partition functions

Let $Z(\omega;\mathbf y)$ be Rayleigh, and let $\widetilde Z$ be its symmetrization. For each $e\in E$ and each pair $\{e,f\}\subseteq E$, use the auxiliary quantities $z(e)$ and $z(ef)$ defined from the symmetrized derivatives in the source. Averaged symmetrization inequality. For every $\mathbf y>\boldsymbol 0$,

$$\sum_{\{e,f\}\subseteq E}\bigl(z(e)z(f)-z(ef)\widetilde Z^{12}\bigr)\geq0.$$

The inequality is presented as a sufficient condition for the symmetrization conjecture and is not proved in the paper.