The potential of the quiver associated to surfaces is the canonical one given by [Labardini-Fragoso's 2009 paper](https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/pdn051), who proved that the the QP associated to surfaces whose boundary is nonempty is rigid hence non-degenerate, in the sense of [this paper](https://link.springer.com/article/10.1007/s00029-008-0057-9). The author conjectured that the QP associated to surfaces with empty boundary is non-rigid but still non-degenerate. Later [Sefi Ladkani](http://arxiv.org/abs/1207.3778v1) proved that the QP associated to surfaces with empty boundary is non-rigid.


So I wonder how much is done on the non-degeneracy? The case of once punctured torus is easy: However you mutate, you still get the same QP. What is known for other closed surfaces?



How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?
In [On cluster algebras from once punctured closed surfaces](https://arxiv.org/abs/1310.4454) another paper from Sefi Ladkani, it is shown in the theorem whose statement begins at the bottom of page 1 that if $Q$ is the adjacency quiver of a once punctured closed surface (so the once punctured torus as well as higher genus), then there is a nondegenerate potential on $Q$ (see part (e)).


