The oriented-percolation critical crossing summability conjecture

Fix $\ell_0>0$ and let ${\cal R}_i:=\Lambda_{\ell_i,\ell_i/12}$, where $\ell_i=2\ell_{i-1}$. Let $\partial R_3^i$ and $\partial R_4^i$ be the two short sides of ${\cal R}_i$, parallel to $-e_1+e_2$, and let $S_i$ be the event that ${\cal R}_i$ is crossed in the parallel direction. Let $\mu^\rho$ denote Bernoulli measure and let $p_c^{OP}$ be the critical probability for oriented site percolation.

Oriented-percolation critical crossing summability conjecture.

$$\sum_{i=1}^{\infty}\left|\log\bigl(\mu^\rho(S_i)\bigr)\right|<\infty\quad\text{at }\rho=p_c^{OP}.$$

This is a milder version of the anisotropic crossing conjecture, introduced because it is sufficient to prove discontinuity of the transition. The source does not provide a rigorous proof or a resolution.