Simple product conjecture for dual canonical basis elements

Let $\mathscr B(\infty)$ be the crystal indexing the dual canonical basis. For basis elements $G^{\operatorname{up}}(b_i)$, write $b_i\bot b_j$ for the compatibility relation used in the source, and let $q^{\mathbb Z}{\mathbf B}^{\operatorname{up}}$ denote the dual canonical basis up to powers of $q$. Simple product conjecture. For any sequence $b_1,\ldots,b_l\in\mathscr B(\infty)$, the following are equivalent: (1) $G^{\operatorname{up}}(b_1)\cdots G^{\operatorname{up}}(b_l)\in q^{\mathbb Z}{\mathbf B}^{\operatorname{up}}$; (2) $G^{\operatorname{up}}(b_i)G^{\operatorname{up}}(b_j)\in q^{\mathbb Z}{\mathbf B}^{\operatorname{up}}$ for every $i<j$. The claim is motivated by monoidal categorification; one implication is stated to be clear, while the converse is the substantive open direction.