Generalized amalgamation conjecture for irreducible 3-manifolds

Let $M_i$ ($i=1,2$) be irreducible 3-manifolds with connected boundary, with $\partial M_1\cong\partial M_2\cong F$. Let $\mathcal{D}_i$ be the set of essential curves in $F$ that bound disks in $M_i$, and let $\phi:\partial M_1\to\partial M_2$ be a homeomorphism. The distance between $\mathcal{D}_2\cup\mathcal{C}_2$ and $\phi(\mathcal{D}_1\cup\mathcal{C}_1)$ is measured in the curve complex $\mathcal{C}(F)$. Generalized amalgamation conjecture. There are essential curves $\mathcal{C}_i$ ($i=1,2$) in $\partial M_i$ such that, if this distance is sufficiently large, then any minimal-genus Heegaard splitting of $M_1\cup_\phi M_2$ can be constructed from an amalgamation. This would extend the preceding amalgamation result beyond the assumptions that the manifolds are atoroidal and have incompressible boundary; the source gives no evidence that the statement has been resolved.