  We tackle the classification problem of non-degenerate potentials for quivers
arising from triangulations of surfaces in the cases left open by
Geiss-Labardini-Schr\"oer. Namely, for once-punctured closed surfaces of
positive genus, we show that the quiver of any triangulation admits infinitely
many non-degenerate potentials that are pairwise not weakly right-equivalent;
we do so by showing that the potentials obtained by adding the 3-cycles coming
from triangles and a fixed power of the cycle surrounding the puncture are well
behaved under flips and QP-mutations. For twice-punctured closed surfaces of
positive genus, we prove that the quiver of any triangulation admits exactly
one non-degenerate potential up to weak right-equivalence, thus confirming the
veracity of a conjecture of the aforementioned authors.
