The compatibility conjecture for real modules

Let $\mathcal{A}$ be as above, let $M_1,M_2\in\mathcal{P}_\ell^+$ be dominant monomials such that $L(M_1)$ and $L(M_2)$ are real, and let $\mathbf{g}_{M_1},\mathbf{g}_{M_2}$ be the corresponding $\mathbf{g}$-vectors. Compatibility conjecture.

$$\chi_q(L(M_1))\chi_q(L(M_2))=\chi_q(L(M_1M_2))$$

if and only if

$$\mathfrak{e}(\mathbf{g}_{M_1},\mathbf{g}_{M_2})=0.$$

This is presented as the monoidal analogue of compatibility of cluster monomials; the source does not state a resolution.