Berenstein–Zelevinsky's multiplicativity conjecture for dual canonical basis vectors

Let $Q$ be a Dynkin quiver of type $cA_n$, $cD_n$ or $cE_{6,7,8}$, let $U_v^-$ be the negative part of the quantized enveloping algebra of the corresponding Lie algebra, and let ${cmathcal B}^*$ be its dual canonical basis. For $C,D\in {cmathcal B}^*$, call $C$ and $D$ multiplicative if their product is a scalar multiple of a single dual canonical basis vector, and quasi-commutative if $CD=\lambda DC$ for some $\lambda\in\mathbb{Q}(v)$. Berenstein–Zelevinsky's conjecture. Two dual canonical basis vectors are multiplicative if and only if they are quasi-commutative. One direction was proved by Reineke, but Leclerc constructed quasi-commutative elements that are not multiplicative, so the conjecture is refuted.