Heegaard genus lower bound for gluing 3-manifolds

Let $M_1$ and $M_2$ be compact orientable $3$-manifolds with incompressible boundary components $F_1$ and $F_2$, respectively, and let $h:F_1\rightarrow F_2$ be any homeomorphism. Write $g(M)$ for the Heegaard genus of $M$. Gluing Heegaard genus conjecture.

$$g(M_1\cup_h M_2)\geq \max\{g(M_1),g(M_2)\}-g(F_1).$$

This is presented as a much stronger possible statement than the preceding knot connected-sum conjecture, with less evidence or hope for a proof. The source gives no resolution, so its status remains open.