A counter example was found on the nine holed torus, and I have heard there is now one on the five holed torus as well so the conjecture is false in general. However what is still not known is for which n does the conjecture hold for the n-holed torus, and in particular the one holed torus is always of interest and remains open.

## False in general

A counter example was found on the nine holed torus, and I have heard there is now one on the five holed torus as well so the conjecture is false in general. However what is still not known is for which n does the conjecture hold for the n-holed torus, and in particular the one holed torus is always of interest and remains open.