The pairwise symmetrization inequality for Rayleigh partition functions

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

$$z(e)z(f)-z(ef)\widetilde Z^{12}\geq0.$$

This is described as a simpler and stronger sufficient conjecture for the preceding averaged inequality, hence for the symmetrization conjecture; it remains open.