The high-complexity torus-gluing conjecture for unstabilized Heegaard splittings

Let $X$ and $Y$ be 3-manifolds, each with a single incompressible torus boundary component, and let $M$ be obtained by gluing them along their boundaries. There is a complexity on maps $\varphi \colon \partial X \to \partial Y$ and an integer $n(X,Y)$ such that, when the complexity of $\varphi$ is greater than $n(X,Y)$, the amalgamation of any unstabilized Heegaard splittings of $X$ and $Y$ is unstabilized. High-complexity torus-gluing conjecture. The preceding assertion holds. The conjecture proposes that sufficiently complicated torus gluings prevent stabilization in amalgamated splittings; the supplied source gives no resolution.