Cluster structure conjecture for finite-type quantum toroidal modules

Let $\mathcal{O}^{fin}_{{\mathcal B}^\perp}$ be the category of finite-type modules, let $R=\operatorname{Rep}\,\mathcal{O}^{fin}_{{\mathcal B}^\perp}$ be its Grothendieck ring, and let $\Gamma$ be the quiver with vertex set $\Gamma_0=\mathbb{Z}^3$ and arrows

$$(i_1,j_1,k_1)\to(i_2,j_2,k_2)$$

exactly when $(i_2,j_2,k_2)$ is one of $(i_1+1,j_1,k_1)$, $(i_1,j_1+1,k_1)$, or $(i_1,j_1,k_1+1)$. For $a\in\mathbb{C}^\times$, set $q^{(i,j,k)}=q_3^iq_1^jq_2^k$ and define the initial cluster by $c_0(i,j,k)=[M^+(aq^{(i,j,k)})]$. A module is prime if it cannot be written as a tensor product of two non-one-dimensional modules, real if its tensor square is irreducible, and normalized if $V_0\neq0$ and $V_n=0$ for all $n>0$. Cluster structure conjecture. (i) The cluster $(\Gamma,c_0)$ can be repeatedly mutated in any sequence of directions, with each mutation corresponding to a short exact sequence in $\mathcal{O}^{fin}_{{\mathcal B}^\perp}$. (ii) Every cluster variable obtained through mutations is an irreducible prime real module. (iii) Every irreducible prime real normalized module appears as a cluster variable. (iv) Any tensor product of modules corresponding to variables in the same cluster is irreducible; in particular, both terms in the left-hand side of the mutation equation correspond to irreducible modules. This conjecture proposes a cluster-algebraic organization of finite-type modules, but the supplied text gives no resolution status.