Gordon's stabilization conjecture for connected sums of Heegaard splittings

Let $M_1$ and $M_2$ be closed, orientable $3$-manifolds, and let $H_i$ be a Heegaard surface in $M_i$ for $i=1,2$. The connected sum $H_1\# H_2$ is a Heegaard surface in $M_1\# M_2$; a Heegaard surface is stabilized if it comes from stabilizing a lower-genus Heegaard surface. Gordon's conjecture. If $H_1\# H_2$ is a stabilized Heegaard surface in $M_1\# M_2$, then either $H_1$ or $H_2$ is stabilized. This conjecture concerns how stabilization behaves under connected sum. The supplied source does not establish whether the conjecture was resolved, so its status is recorded as open.