Path extremal conjecture for zero forcing sets of fixed size

Let $G$ be an $n$-vertex graph, and let $z(G;k)$ denote the number of zero forcing sets of $G$ having size $k$. Let $P_n$ be the path on $n$ vertices. Fixed-size zero forcing conjecture. If $G$ is an $n$-vertex graph, then for all $k$,

$$z(G;k)\le z(P_n;k)=\binom{n}{k}-\binom{n-k-1}{k}.$$

This conjecture would imply the path extremal conjecture for the probability that a random vertex set is zero forcing, via the expansion of that probability in terms of the numbers $z(G;k)$. The paper presents it as a stronger conjecture and does not establish it.