Berenstein–Zelevinsky congruence conjecture for quasi-commuting dual canonical basis elements

Let ${\cal M}$ be the set of multisegments. For ${\bf m},{\bf n}\in{\cal M}$, let $G^*({\bf m})$ and $G^*({\bf n})$ be the corresponding dual canonical basis elements, let $b({\bf m},{\bf n})$ be the bilinear-form exponent appearing in the source, let $E^*({\bf m}+{\bf n})$ be the corresponding monomial basis element, and let ${\cal L}^*$ be the dual canonical lattice. Assume that $G^*({\bf m})$ and $G^*({\bf n})$ quasi-commute. Berenstein–Zelevinsky's congruence conjecture. Then

$$v^{b({\bf m},{\bf n})}G^*({\bf m})G^*({\bf n})\equiv E^*({\bf m}+{\bf n})\pmod {v{\cal L}^*}.$$

This is presented in the source as a reduction of the multiplicative conjecture: it specifies the leading term modulo $v{\cal L}^*$ of the normalized product. Its resolution is not established by the supplied context.