The bounded-genus high-complexity conjecture for higher-genus gluings

Let $X$ and $Y$ each have a single incompressible boundary component of genus at least two, let $\varphi\colon\partial X\to\partial Y$ be a gluing map, and let $M=X\cup_\varphi Y$. There is a complexity on such maps and an integer $n(X,Y,g)$ such that, if the complexity of $\varphi$ is greater than $n(X,Y,g)$, the amalgamation of any unstabilized Heegaard splittings of $X$ and $Y$ whose genera are less than $g$ is unstabilized. Bounded-genus higher-genus gluing conjecture. The preceding assertion holds. This is a bounded-genus refinement of the analogous torus-gluing claim: sufficiently complicated gluings are expected to preserve unstabilized amalgamations only for splittings below a genus bound; the supplied source gives no resolution.