Euler characteristics and actions of automorphism groups of free groups

Let $M^{r}$ be a connected orientable manifold with the Euler characteristic $\chi(M)\not \equiv 0\operatorname{mod}6$. Denote by $\mathrm{SAut}(F_{n})$ the unique subgroup of index two in the automorphism group of a free group. Then any group action of $\mathrm{SAut}(F_{n})$ (and thus the special linear group $\mathrm{SL}_{n}(\mathbb{Z})$) $(n\geq r+2$) on $M^{r}$ by homeomorphisms is trivial. This confirms a conjecture related to Zimmer's program for these manifolds.