Stable rational abelianization conjecture for the handlebody hyperelliptic Torelli group

Let $\mathcal{H}_B\mathcal{I}_{g,p}^b$ denote the handlebody hyperelliptic Torelli group, and let $\overline{W}_{\mathbb Q}$ and $W_{\mathbb Q}$ be the rational modules appearing in its Johnson-type homomorphism description. Let $\Theta^*$ be the induced map on second rational cohomology. Stable rational abelianization conjecture. For $g \gg 0$, we have

$$H_1(\mathcal{H}_B\mathcal{I}_{g,p}^b; \mathbb Q) \cong \begin{cases} \overline{W}_{\mathbb Q} & (p=b=0) \\ W_{\mathbb Q} & (p+b = 1) \end{cases}.$$

Together with the preceding conjecture concerning the modules in $\operatorname{ker}(\Theta^*)$, this would imply that the known theorem describes the entire cup-product map. The stable abelianization statement remains unproved in the source.