The generalized cluster algebra conjecture for the Grothendieck ring of $\mathcal{C}_{\varepsilon^\mathbb{Z}}$

Let $l\geq 2$, and let $\mathcal{C}_{\varepsilon^\mathbb{Z}}$ be the category whose Grothendieck ring is under consideration. Write $\mathcal G_l$ for a generalized cluster algebra of rank $2l-2$, with coefficients $\lambda_i=\lbrack L(\mathbf{Y}_{i})\rbrack$ for $i=1,2$, where

$$\mathbf{Y}_1=Y_{1,0}Y_{1,2}\dots Y_{1,2l-2},\qquad \mathbf{Y}_2=Y_{2,1}Y_{2,3}\dots Y_{2,2l-1}.$$

Its initial exchange polynomials are $\theta_r^0(u,v)=u+v$ for $r\in\llbracket 1,2l-3\rrbracket$ and $\theta_{2l-2}^0(u,v)=u^3+\lambda_1u^2v+\lambda_2uv^2+v^3$; the initial cluster variables are the classes of the simple modules listed in the claim below. The generalized cluster algebra conjecture. The Grothendieck ring of $\mathcal{C}_{\varepsilon^\mathbb{Z}}$ is isomorphic to $\mathcal G_l$, with cluster variables corresponding to

$$x_{2k+1}=\lbrack L(Y_{1,0}Y_{1,2l-2}\dots Y_{1,2l-2k})\rbrack\quad(k\in\llbracket 0,l-2\rrbracket),$$ $$x_{2k}=\lbrack L(Y_{2,2l-1}Y_{2,2l-3}\dots Y_{2,2l-2k+1})\rbrack\quad(k\in\llbracket 1,l-2\rrbracket),$$ $$x_{2l-2}=\lbrack L(Y_{1,0}Y_{1,2l-2}Y_{1,2l-4}\dots Y_{1,4}Y_{2,2l-1}Y_{2,2l-3}Y_{2,2l-5}\dots Y_{2,5}Y_{2,3})\rbrack,$$

and the generalized cluster monomials are mapped to classes of simple modules. This conjecture proposes a cluster-algebraic realization of the Grothendieck ring and a parametrization of simple-module classes by generalized cluster monomials; the conjectural identification is based on computations in the $A_2$ case and is not established in the supplied text.