Gleitz's generalized cluster categorification conjecture for restricted quantum loop algebra modules

Let $cal C_{\varepsilon,xi}$ be the full subcategory of finite-dimensional modules of the restricted quantum loop algebra $U_\varepsilon^{{\rm res}}(Lmathfrak{sl}_k)$ whose composition factors are $L(m)$ with $m$ in the bipartition-dependent monoid $cal P^+_{\varepsilon,xi}$. For $k=3$ and $ell\tsim\bb Z_{\geq 2}$, Gleitz's conjecture asserts that the Grothendieck ring $K_0(cal C_{\varepsilon,xi})$ is isomorphic to a generalized cluster algebra of rank $2ell-2$, and that generalized cluster monomials correspond to classes of simple modules. This extends the cases in which Gleitz established generalized cluster algebra structures of types $C_{ell-1}$ for $k=2$ and $G_2$ for $k=3$, $ell=2$; the general $k=3$ case remains conjectural.