  Berenstein and Zelevinsky introduced quantum cluster algebras [Adv. Math,
2005] and the triangular bases [IMRN, 2014]. The support conjecture by
Lee-Li-Rupel-Zelevinsky [PNAS, 2014] asserts that the support of a triangular
basis element for a rank-2 cluster algebra is bounded by an explicitly
described region that is possibly concave. In this paper, we prove the support
conjecture for all skew-symmetric rank-2 cluster algebras.
