Critical percolation threshold has zero percolation probability

Determine whether the Bernoulli site-percolation probability at the critical threshold satisfies \(\theta_{\mathbb{Z}^d}(p_c(\mathbb{Z}^d))=0\) for every dimension \(d\ge 2\).

Background

The paper relates the survival of the branching random walk with hard killing to the geometry of the open cluster containing the origin in Bernoulli site percolation on $\mathbb{Z}^d$. The quantity $\theta_{\mathbb{Z}^d}(p)$ denotes the probability that the origin belongs to an infinite open cluster, and $p_c(\mathbb{Z}^d)$ is the percolation threshold.

Whether $\theta_{\mathbb{Z}^d}(p_c(\mathbb{Z}^d))=0$ is known in dimension two and for dimensions $d\ge 11$, but remains unresolved in the intervening dimensions. The problem is a classical percolation conjecture explicitly identified by the paper.