Stable integral abelianization conjecture for handlebody Torelli groups

Let $\mathcal{HI}_{g,p}^b$ be the handlebody Torelli group, and let $\tau(\mathcal{HI}_{g,p}^b)$ and $\sigma(\mathcal{HK}_{g,p}^b)$ denote the images of the Johnson and Birman--Craggs--Johnson homomorphisms, respectively. Stable integral abelianization conjecture. For $g \gg 0$, we have

$$H_1(\mathcal{HI}_{g,p}^b; \mathbb Z) \cong \tau(\mathcal{HI}_{g,p}^b) \oplus \sigma(\mathcal{HK}_{g,p}^b).$$

The conjecture concerns torsion in the integral group homology, which is not addressed by the rational results in the paper; the stable integral decomposition remains open.