Injectivity conjecture for odd-dimensional modular-form classes

Let $\mathcal{M}_{2k+2}$ be the space of modular forms of weight $2k+2$. The construction in Section~ gives a map

$$\mathcal{M}_{2k+2}\longrightarrow H_{4k+3}\bigl(\operatorname{Out}(F_{2k+4})\bigr).$$

Odd-dimensional injectivity conjecture. This map is injective for all $k$. The target lies in codimension $2$ because the virtual cohomological dimension of $\operatorname{Out}(F_{2k+4})$ is $4k+5$; the conjecture would imply at least linear growth of the corresponding homology dimensions for even rank.