Product-region classification conjecture for topologically minimal surfaces

Let $F$ be a surface of positive genus, and let $I$ denote the closed interval. A surface is topologically minimal when its disk complex is empty or non-contractible. Product-region classification conjecture. The only connected topologically minimal surfaces in $F\times I$ are a single copy of $F$ and two copies of $F$ connected by an unknotted tube.

The source explains that the conjecture would reduce the classification in $F\times I$ to the known incompressible and strongly irreducible cases, and, assuming the conjecture for $S^3$, would be equivalent to saying that $F\times I$ has no topologically minimal surfaces of index greater than one.