Rank-two conjecture for groups even on maximal proper subsets

Let $[m]$ and $[n']$ be the two underlying sets of sizes $m$ and $n$, and let $\Gamma_{m\oplus n}$ denote the group of permutations that are even on every maximal proper subset of their disjoint union. For a group $G$, write $\operatorname{rank}(G)$ for its minimum number of generators.

Rank-two conjecture. If $m\geqslant n\geqslant 2$ and

$$(m,n)\notin\{(2,2),(3,3),(4,3),(4,4)\},$$

then

$$\operatorname{rank}(\Gamma_{m\oplus n})=2.$$

The conjecture is motivated by extensive GAP computations: the four excluded cases include the exceptional small cases, while the other checked examples have rank two. The claim is left open.