The high-distance handlebody-gluing conjecture for Heegaard genus

Let $X$ have a single boundary component $F$ with $g(F)\geq 2$, and let $V\cup_H W$ be a minimal-genus Heegaard splitting of $X$ with $F$ contained in the compression body $V$. Let $\mathcal W$ be the set of vertices of the curve complex of $H$ corresponding to boundaries of disks in $W$. For each loop $\alpha$ on $F$, choose an essential annulus in $V$ meeting $F$ in $\alpha$ and $H$ in a loop $\alpha_H$. After gluing a handlebody $\mathcal H(F)$ to $\partial X$, let $\mathcal V_F$ consist of those vertices $\alpha_H$ for which $\alpha$ bounds a disk in $\mathcal H(F)$. High-distance handlebody-gluing conjecture. If the distance between $\mathcal W$ and $\mathcal V_F$ is large enough, then $H$ is a minimal-genus Heegaard splitting of $X\cup_F\mathcal H(F)$. This conjecture is proposed as an answer to Lackenby's question about degeneration of Heegaard genus under handlebody gluing; the supplied source gives no resolution.