Dual canonical basis conjecture for acyclic quantum cluster algebras

Let $\mathcal A^q$ be a quantum cluster algebra associated with an acyclic quiver and a $z$-coefficient pattern. Let $C$ denote Berenstein–Zelevinsky's triangular basis, and let the dual canonical basis be the basis inherited from the corresponding quantum unipotent subgroup. Dual canonical basis conjecture. The dual canonical basis agrees with Berenstein–Zelevinsky's triangular basis $C$. This conjecture would imply that Berenstein–Zelevinsky's triangular basis contains all quantum cluster monomials. It is proved in the paper's main result for quantum cluster algebras arising from acyclic skew-symmetric matrices, while the stated acyclic-quiver formulation is presented as a conjecture.