Sedgwick's conjecture on infinitely many irreducible Heegaard splittings

Let $M$ be a closed, orientable $3$-manifold. An irreducible Heegaard splitting is a Heegaard splitting with no reducing pair of compressing disks, and two splittings are pairwise non-isotopic if no two are related by an ambient isotopy.

Sedgwick's conjecture. If $M$ contains infinitely many irreducible Heegaard splittings that are pairwise non-isotopic, then $M$ is Haken.

This is the stronger conjecture from which the paper separates the fixed-genus and increasing-genus cases. The source presents it as conjectural and attributes it to Sedgwick.