The conjecture identifying the band basis with the common triangular basis

Let the skein algebra of a marked surface have frozen variables, and consider its localization at those frozen variables. The skein algebra carries a (quantized) band basis and a common triangular basis, the latter being an analogue of the dual canonical basis. Band–triangular basis conjecture. The (quantized) band basis is the common triangular basis of the skein algebra after localization at the frozen variables. If true, this conjecture implies Thurston's conjecture because the common triangular basis provides an analogue of the dual canonical basis. Its status is not established in the supplied source.