Path extremal conjecture for random set zero forcing

Let $G$ be an $n$-vertex graph, let $P_n$ denote the path on $n$ vertices, and let $B_p(G)$ be the random subset of vertices obtained by including each vertex independently with probability $p$, where $0\le p\le 1$. Write $\operatorname{ZFS}(G)$ for the family of zero forcing sets of $G$. Path extremal conjecture. If $G$ is an $n$-vertex graph and $0\le p\le 1$, then

$$\Pr[B_p(G) \in \operatorname{ZFS}(G)]\le \Pr[B_p(P_n) \in \operatorname{ZFS}(P_n)],$$

with equality holding if and only if either $p\in \\{0,1\\}$ or $G=P_n$. This is the random-set analogue of the classical fact that the path has the smallest zero forcing number among $n$-vertex graphs. The paper proves the inequality, with the stated equality conditions, for trees when $n$ is sufficiently large, and proves a constant-factor comparison for general graphs.