Two-set p-capacity factorisation conjecture for the IIC

Let $A,B\subseteq\mathbb Z^d$ be finite, let $z+B=\{z+b:b\in B\}$, let $\mathcal C_\infty(z)$ be the Incipient Infinite Cluster rooted at $z$, and let

$$\tau_\infty(z)=\mathbb P(z\in\mathcal C_\infty(0)).$$

Two-set p-capacity factorisation conjecture.

$$\lim_{\|z\|\to\infty}\frac{\operatorname{pCap}(A)+\operatorname{pCap}(B)-\operatorname{pCap}(A\cup(z+B))}{\tau_\infty(z)}=\operatorname{pCap}(A)\operatorname{pCap}(B).$$

This is presented as a stronger form of the IIC hitting conjecture. The corresponding result is available for branching capacity and general Bessel–Riesz capacities, but the percolation statement is not proved in the supplied text.