Rubinstein's conjecture on strongly irreducible Heegaard splittings

Let $(M,g)$ be a closed oriented Riemannian $3$-manifold, and let $H$ be a strongly irreducible Heegaard splitting of $M$.

Rubinstein's conjecture. $H$ is either isotopic to a minimal surface or isotopic to the oriented double cover of a non-orientable minimal surface with a vertical handle attached.

This conjecture concerns realizing strongly irreducible Heegaard splittings by minimal surfaces. The source states that Rubinstein announced this result and sketches a proof, but that a complete proof had apparently not been published; the conjecture is presented here as an open statement.