The exchange-pair conjecture for real prime modules

Let $M$ and $N$ be dominant monomials such that $L(M)$ and $L(N)$ are real prime modules, and let $\mathfrak{e}(\mathbf{g}_M,\mathbf{g}_N)$ be their generic symmetrized $E$-invariant. Exchange-pair conjecture. There exist distinct dominant monomials $U\neq V$ such that

$$\chi_q(L(M))\chi_q(L(N))=\chi_q(L(U))+\chi_q(L(V))$$

if and only if

$$\mathfrak{e}(\mathbf{g}_M,\mathbf{g}_N)=1.$$

The statement is the expected monoidal counterpart of the exchange-pair criterion; no resolution is supplied in the source.