Finite-type Grothendieck-ring generation conjecture for quantum toroidal $\mathfrak{gl}(1)$

Let $\mathcal{O}^{fin}_{{\mathcal B}^\perp}$ be the category of finite-type modules, and let $K_0(\mathcal{O}^{fin}_{{\mathcal B}^\perp})$ denote its completed Grothendieck ring. For $i,j,k\in\mathbb{Z}_{>0}$ and $u\in\mathbb{C}^\times$, define

$$N_{i,j,k}^+(u)=L\bigl(\Psi_{N_{i,j,k}^+(u)}\bigr),\qquad \Psi_{N_{i,j,k}^+(u)}(z)=\frac{(1-q_3^{-i}u/z)(1-q_1^{-j}u/z)(1-q_2^{-k}u/z)}{1-u/z}.$$

Also let $M^+(u)$ be the positive fundamental module. Finite-type Grothendieck-ring generation conjecture. The completed Grothendieck ring of $\mathcal{O}^{fin}_{{\mathcal B}^\perp}$ is topologically generated by $[N^+_{i,j,k}(u)]$ with $i,j,k\in\mathbb{Z}_{>0}$, $u\in\mathbb{C}^\times$, and $[M^+(u)]$ with $u\in\mathbb{C}^\times$. This predicts a generating family for the representation ring of finite-type modules; the supplied text gives no resolution status.