Minimal layered-triangulation conjecture for handlebodies

Let $H_g$ be a genus-$g$ handlebody, and let $\tau$ be a one-vertex triangulation of its boundary. A $[\tau]$-layered-triangulation is a layered-triangulation of $H_g$ extending $\tau$, and a minimal $[\tau]$-layered-triangulation is one with the fewest tetrahedra among all such extensions. Minimal layered-triangulation conjecture for handlebodies. The minimal triangulation extending the one-vertex triangulation $\tau$ on the boundary of a genus-$g$ handlebody is a minimal $[\tau]$-layered-triangulation. This is presented as an analogue of the solid-torus conjecture: it asks whether the layered extension realizes the minimum number of tetrahedra for each prescribed one-vertex boundary triangulation. The supplied text gives examples in genus two but no resolution of the general claim.