Critical surfaces can be isotoped to index 2 minimal surfaces

A critical surface is a Heegaard surface with the criticality property defined in the paper, and an index 2 minimal surface is a minimal surface whose second variation has index 2. Critical-surface minimality conjecture. Any critical surface can be isotoped to be an index 2 minimal surface. This conjecture proposes that the correspondence between incompressible and strongly irreducible Heegaard surfaces and index 0 and 1 minimal surfaces extends to critical Heegaard surfaces and index 2 minimal surfaces. The source provides motivation from known results for incompressible and strongly irreducible surfaces but does not state a resolution.