Waldhausen-type conjecture for trisections of $S^4$

A trisection of a closed $4$-manifold is a decomposition into three $4$-dimensional handlebodies meeting pairwise in $3$-dimensional handlebodies and all together in a closed surface. The genus $0$ trisection is the standard trisection of $S^4$, and a stabilization is the standard operation increasing its genus.

Waldhausen-type conjecture. Every trisection of $S^4$ is isotopic to either the genus $0$ trisection or its stabilization.

This is presented as a $4$-dimensional analogue of Waldhausen's theorem on Heegaard splittings. The source gives no resolution status.