Conjecture on irreducible Heegaard splittings of arbitrarily large genus

Let $M$ be a closed, orientable $3$-manifold. An irreducible Heegaard splitting is a Heegaard splitting with no reducing pair of compressing disks, and its genus is the genus of its Heegaard surface.

Large-genus Heegaard splitting conjecture. If $M$ contains irreducible Heegaard splittings of arbitrarily large genus, then $M$ is Haken.

The paper identifies this as the increasing-genus half of Sedgwick's stronger conjecture and says it was the inspiration for the work. The source gives no evidence that it had been resolved.