Topological minimality exclusion conjecture for the 3-sphere and 3-ball

Let $M$ be either the 3-sphere $S^3$ or the 3-ball $B^3$, and let a topologically minimal surface mean a surface whose disk complex is empty or non-contractible. Topological minimality exclusion conjecture. $S^3$ and $B^3$ do not contain topologically minimal surfaces.

A corollary would be that handlebodies contain no closed topologically minimal surfaces; the conjecture also rules out the third conclusion of the stated corollary on topologically minimal Heegaard surfaces.