Generation conjecture for the handlebody Johnson kernel

Let $\mathcal{H}_g$ be the handlebody group, let $\mathcal{K}_g$ be the Johnson kernel of the mapping class group, and define the handlebody Johnson kernel by

$$\mathcal{HK}_g:=\mathcal{H}_g\cap\mathcal{K}_g.$$

A separating meridian twist is a Dehn twist about a separating meridian of the handlebody. Generation conjecture. The handlebody Johnson kernel $\mathcal{HK}_g$ is generated by separating meridian twists. This would identify the subgroup used in the preceding finite-abelianization theorem with the full handlebody Johnson kernel, and would clarify the structure of this infinite-index subgroup of the handlebody Torelli group. The source provides no resolution of the conjecture.