Finiteness conjecture for full Heegaard splittings

Let $M$ be a closed, orientable three-manifold. A Heegaard splitting is full when it does not have the disjoint curve property in the terminology of the source. Finiteness conjecture for full Heegaard splittings. There are only finitely many full Heegaard splittings of $M$, up to isotopy.

The paper derives this claim conditionally from the generalized Waldhausen conjecture together with its theorem that sufficiently high-genus splittings have the disjoint curve property. Its status is therefore unresolved in the source.