The first homology conjecture for the handlebody Torelli group

Let $g$ be sufficiently large, let $\mathcal{HI}_g^1$ denote the handlebody Torelli group, and let $B_3^{b_i}$ be the subspace defined from the Birman–Craggs–Johnson homomorphism. Handlebody Torelli homology conjecture. For $g \gg 0$,

$$H_1(\mathcal{HI}_g^1;\mathbb F_2)\cong B_3^{b_i}.$$

This conjecture predicts that the first mod-$2$ homology of the handlebody Torelli group is identified with the subspace arising as the image of the Birman–Craggs–Johnson homomorphism. It is presented as an analogue of the corresponding result for the full Torelli group, but its status is otherwise unresolved here.