Geometric minimality conjecture for topological-index surfaces

Let $H$ be a surface with topological index $n$. A surface is geometrically minimal if it is minimal with respect to the geometric notion used in the paper. Geometric minimality conjecture. The surface $H$ is isotopic to a geometrically minimal surface whose index is at most $n$.

This would relate the disk-complex notion of topological index to geometric minimal-surface theory. The supplied text gives no resolution status or further evidence, so the conjecture remains open in this record.