Bachman's index-two minimal surface conjecture

Let $M$ be a non-Haken $3$-manifold, and let $a3_1$ and $a3_2$ be strongly irreducible Heegaard surfaces in $M$ that are not isotopic. A lowest genus stabilization is a common stabilization of $a3_1$ and $a3_2$ having minimal genus among such stabilizations. Bachman's index-two minimal surface conjecture. There exists an index $2$ minimal surface whose genus equals the genus of a lowest genus stabilization of $a3_1$ and $a3_2$. This conjecture proposes that the two-parameter family obtained by stabilizing distinct strongly irreducible Heegaard splittings produces an index-two minimal surface with optimal controlled genus in a non-Haken manifold.