The monoidal–additive correspondence conjecture

Let $\mathcal{A}$ be a cluster algebra with monoidal categorification $\mathscr{C}_\ell$ and additive categorification $(\mathcal{F},\widehat{T})$, let $\mathcal{C}$ be the stable category of $\mathcal{F}$, and let $M\in\mathcal{P}_\ell^+$ be dominant. Write $L(M)$ for the associated simple module, $\mathbf{g}_M\in\mathbb{Z}^n$ for the $\mathbf{g}$-vector of $\chi_q(L(M))$, and $\mathfrak{e}(\mathbf{g}_M)$ for the generic degenerate $E$-invariant. Monoidal–additive correspondence conjecture.

$$L(M)\text{ is real}\quad\Longleftrightarrow\quad\mathfrak{e}(\mathbf{g}_M)=0.$$

The source notes that the forward implication follows for reachable monomials, while the full equivalence is used to relate monoidal and additive reachability and remains conjectural.