Fock–Goncharov positivity conjecture for surface skein algebras

Let $\mathord{\mathbb{S}}$ be an orientable surface, and let the multicurve basis be the basis of the surface skein algebra given by multicurves. Applying Chebyshev polynomials of the second kind to this basis gives the associated skein basis. Fock–Goncharov positivity conjecture. The skein basis obtained from the multicurve basis by application of Chebyshev polynomials of the second kind is positive provided

$$\mathord{\mathbb{S}}\neq \mathbb{S}^1\times\mathbb{S}^1.$$

This is a positivity conjecture for surface skein algebras, motivated by the Fock–Goncharov perspective and by the categorification of skein theory. Positivity is known at $q=1$, and at generic $q$ for the torus, the punctured torus, and the 4-punctured sphere; the general case remains open.