Negative association conjecture for Rayleigh measures

Let $Z(\omega;\mathbf y)$ be a partition function on a finite ground set $E$, and let $\mathsf A_1,\mathsf A_2$ be increasing events with disjoint support. Negative association conjecture. If $Z(\omega;\mathbf y)$ is Rayleigh, then for every $\mathbf y>\boldsymbol 0$,

$$\left\langle\boldsymbol 1_{\mathsf A_1\cap\mathsf A_2}\right\rangle\leq \left\langle\boldsymbol 1_{\mathsf A_1}\right\rangle\left\langle\boldsymbol 1_{\mathsf A_2}\right\rangle.$$

The claim says that the Rayleigh pairwise negative-correlation property implies negative association; the paper records a theorem for homogeneous Rayleigh partition functions, but not this general conjecture.