Filling homotopy conjecture for regularly homotopic filling immersions

Let $S$ be an arbitrary surface and let $f,g:S\rightarrow M$ be filling immersions that are regularly homotopic.

Filling homotopy conjecture. Regularly homotopic filling immersions of arbitrary surfaces are filling homotopic.

The preceding theorem proves the assertion for nulhomotopic filling immersions of the two-sphere, while the extension to arbitrary surfaces is left as a conjecture.