The maximal product-of-trees embedding conjecture for handlebody groups

Let $F_2$ be the free group of rank $2$, let $F_2^n$ denote its $n$-fold direct product, and let $\mathcal{H}_g$ be the handlebody group of genus $g$. A subgroup or map is quasi-isometrically embedded when the induced orbit map, equivalently the inclusion with respect to word metrics, is a quasi-isometric embedding. Maximal embedding conjecture. The maximal integer $n$ such that $F_2^n$ quasi-isometrically embeds into $\mathcal{H}_g$ is

$$n=g-1.$$

This asks for the largest number of independent free-group factors that can occur in a quasi-isometric embedding into the handlebody group. The surrounding discussion identifies this as a main question arising from the paper's work; no resolution is stated here.