The extended g-vector classification conjecture for real prime modules

Let $\mathcal{Q}^{\leq \xi}_l$ be the principal quiver of $\Gamma^{\leq \xi}_l$, let $\mathcal{C}^{\leq \xi}_l$ be its associated cluster category, and let $\mathscr{C}^{\leq \xi}_l$ be the corresponding monoidal categorification. For an indecomposable reachable rigid object $M$, write $\widetilde{\mathbf{g}}(M)$ for its extended $\mathbf{g}$-vector, $\mathbf{g}(M)$ for its $\mathbf{g}$-vector, and $F_M$ for its $F$-polynomial; let $\mathbb{P}$ be the tropical semifield generated by the variables $y_{i,r}$. The extended $\mathbf{g}$-vector classification conjecture. A non-frozen real prime module in $\mathscr{C}^{\leq \xi}_l$ is completely determined by the extended $\mathbf{g}$-vector of the corresponding indecomposable reachable rigid object up to equivalence, and the map

$$\Phi^{\leq \xi}_{l}: \{\text{indecomposable reachable rigid objects in }\mathcal{C}^{\leq \xi}_l\}\longrightarrow\{\text{classes of non-frozen real prime objects in }\mathscr{C}^{\leq \xi}_l\}$$ $$M\longmapsto\left[L\left((\mathbf{z}^{\leq \xi}_l)^{\widetilde{\mathbf{g}}(M)}\right)\right]$$

defines a bijection, where

$$(\mathbf{z}^{\leq \xi}_l)^{\widetilde{\mathbf{g}}(M)}=\frac{(\mathbf{z}^{\leq \xi}_{l-1})^{\mathbf{g}(M)}}{F_M|_{\mathbb{P}}\left((y_{i,r})_{(i,r)\in\widehat{I}^{\leq \xi}_{l-1}}\right)}.$$

This conjecture proposes a classification of non-frozen real prime modules by extended $\mathbf{g}$-vectors. The surrounding results establish the relevant cluster-algebra and categorification correspondences, but the asserted bijection remains open.