The final unstable homology conjecture for general linear groups over $\mathbb{F}_2$

Let $t\in H_1(\operatorname{GL}_2(\mathbb{F}_2);\mathbb{F}_2)$ be the element determined by the matrix

$$\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right).$$

Final unstable homology conjecture. For every $m\geqslant 1$, the group $H_m(\operatorname{GL}_{2m}(\mathbb{F}_2);\mathbb{F}_2)$ is a single copy of $\mathbb{F}_2$ generated by the class $t^m$.

The theorem preceding the conjecture shows that this group is either trivial or a single copy of $\mathbb{F}_2$ generated by $t^m$; the conjecture asserts that the nontrivial possibility always occurs. It would determine the final unstable homology groups for general linear groups over $\mathbb{F}_2$ and, through the maps from automorphism groups of free groups and from integral general linear groups, resolve corresponding ambiguities and show that the known homological stability ranges are sharp.