The homological non-triviality conjecture for actions of special automorphism groups of free groups

Let $M^{k}$ be any closed manifold, and let $F_n$ be a free group of rank $n$. Write $\mathrm{SAut}(F_n)$ for the unique index-two subgroup of $\mathrm{Aut}(F_n)$, and let $\mathrm{SL}_n(\mathbb{Z})$ be the group of integral matrices of determinant one. An action is homologically trivial if it induces the identity on the homology of $M^k$.

Homological non-triviality conjecture. When $n>k+1$, any non-trivial action of $\mathrm{SAut}(F_n)$ or $\mathrm{SL}_n(\mathbb{Z})$ on $M^k$ by homeomorphisms is homologically non-trivial. Equivalently, any homologically trivial action of either group on $M^k$ by homeomorphisms, when $k<n-1$, is trivial.

This refines a folklore conjecture that actions of $\mathrm{Aut}(F_n)$ on compact $k$-manifolds factor through finite groups when $k<n-1$. It is presented as a topological analogue of the Zimmer program; the supplied text gives no resolution of this conjecture.