Generalized Waldhausen conjecture for fixed-genus Heegaard splittings

Let $M$ be a closed, orientable $3$-manifold. A Heegaard splitting is a decomposition of $M$ into two handlebodies meeting along their common boundary surface; it is toroidal when $M$ contains an incompressible torus. The splittings below are pairwise non-isotopic and have the same genus.

Generalized Waldhausen conjecture. If $M$ contains infinitely many Heegaard splittings, pairwise non-isotopic, all of the same genus, then $M$ is toroidal.

The paper describes this as one of the two parts of Sedgwick's stronger conjecture and notes that Jaco and Rubinstein had claimed it, although no manuscript was available when the paper was written.