Equality of (3,3)- and (3,4)-spreading numbers on grid graphs

Let $P_m$ and $P_n$ be paths on $m$ and $n$ vertices, respectively, and let $P_m\square P_n$ denote their Cartesian product. For a graph $G$, write $\sigma_{(p,q)}(G)$ for its $(p,q)$-spreading number. Spreading-number equality conjecture. If $m,n\ge 3$, then

$$\sigma_{(3,3)}(P_m\square P_n)=\sigma_{(3,4)}(P_m\square P_n).$$

The conjecture concerns the unresolved $(3,q)$-spreading numbers for rectangular grid graphs. Since every vertex of $P_m\square P_n$ has degree at most $4$, the $(3,4)$- and $(3,\infty)$-spreading numbers coincide; the conjectured equality would therefore also identify the $(3,3)$-spreading number with the corresponding bootstrap percolation number, whose exact values are not known in general.