Rigid-object and real-simple-module bijection conjectures

Let $\xi$ be an arbitrary height function and let $\ell\geq 2$. Let $Q^{\leq \xi}_\ell$ be the principal quiver, let $A^{\leq \xi}_\ell$ be its Jacobian algebra, and let $\mathcal{C}^{\leq \xi}_\ell$ be the associated cluster category. For $(i,r)\in\widehat{I}^{\leq \xi}_\ell\setminus\widehat{I}^{\leq \xi}_{\ell-1}$, set $f_{i,r}=Y_{i,r}Y_{i,r+2}\cdots Y_{i,\xi(i)}$. A simple module is real if its tensor square is simple and prime if it is not a tensor product of nontrivial modules. Rigid-object bijection conjectures. (1) There is a bijection between reachable indecomposable rigid objects in $\mathcal{C}^{\leq \xi}_\ell$ and real prime simple modules in $\mathscr{C}^{\leq \xi}_\ell$ not of the form $L(f_{i,r})$. (2) There is a bijection between reachable rigid objects in $\mathcal{C}^{\leq \xi}_\ell$ and real simple modules in $\mathscr{C}^{\leq \xi}_\ell$ having no tensor factors of the form $L(f_{i,r})$. These proposed bijections concern the categorification of cluster algebras beyond level one and are not resolved in the source.