The band basis and common triangular basis conjecture

Let $\mathsf{Sk}(\Sigma)$ be the skein algebra of the surface $\Sigma$, and let the band basis be the basis formed by band elements obtained from bangles by replacing each repeated closed loop with the corresponding band curve. Let the common triangular basis be the basis associated with the cluster algebra in the sense of Qin, and allow localization at the frozen variables.

Band–triangular basis conjecture. The (quantized) band basis coincides with the common triangular basis after localization at the frozen variables.

The conjecture concerns the identification of two distinguished bases of a skein or cluster algebra. The source reports only the annular example as known evidence, so the general statement remains open.