The rank-two conjecture for homologically trivial finite group actions on 4-manifolds

Let $M$ be a closed orientable $4$-manifold with Euler characteristic $(M) \neq 0$, and let $G$ be a finite group acting effectively, locally linearly, and homologically trivially on $M$. Here $\operatorname{rank}_p(G)$ denotes the maximum rank of an elementary abelian $p$-subgroup of $G$. Rank-two conjecture. For every odd prime $p$,

$$\operatorname{rank}_p(G) \leq 2.$$

This conjecture proposes a uniform rank restriction for homologically trivial finite group actions on arbitrary closed orientable $4$-manifolds with nonzero Euler characteristic; it is compared in the source with an earlier conjecture of Edmonds. No resolution is indicated in the supplied text.