Gordon's conjecture on stabilization of connected-sum Heegaard splittings

Let $M_+ = \mathcal{V_+} \cup_{S_+} \mathcal{W_+}$ and $M_- = \mathcal{V_-} \cup_{S_-} \mathcal{W_-}$ be Heegaard splittings of closed orientable $3$-manifolds. Form the connected sum Heegaard splitting

$$M=M_+\#M_- = \mathcal{V}\cup_S\mathcal{W},$$

where $S=S_+\#S_-$ and the compression bodies are obtained by boundary connected sum. A Heegaard splitting is stabilized if it admits properly embedded disks on opposite sides whose boundaries intersect in a single point.

Gordon's conjecture. The connected-sum Heegaard splitting $\mathcal{V}\cup_S\mathcal{W}$ is stabilized if and only if at least one of the summand splittings

$$\mathcal{V_+}\cup_{S_+}\mathcal{W_+}\qquad\text{or}\qquad\mathcal{V_-}\cup_{S_-}\mathcal{W_-}$$

is stabilized.

This conjecture concerns whether stabilization can arise from taking a connected sum of Heegaard splittings. The source presents it as Gordon's conjecture, and the supplied parser status is unknown; the statement's resolution should therefore be checked.