Existence of generalized Macmahon modules by analytic continuation

Fix non-negative integers $l_1,\dots,l_t,l'_1,\dots,l'_t$, and for sufficiently large $k$ let $L_k=(l_1,\dots,l_t,0,\dots,0,l'_t,\dots,l'_1)$ be the vector with $k$ components. Let $\mathcal N_{\gamma(L_k),\beta}^{(p)}(u)$ be the corresponding module for the quantum toroidal algebra $\mathcal E_n$. Existence conjecture for generalized Macmahon modules. There exists a lowest weight admissible tame $\mathcal E_n$-module $\mathcal M_{\gamma(l),\beta}^{(n;p),\gamma(l')}(u,K)$ of level $K$ that is the analytic continuation of $\mathcal N_{\gamma(L_k),\beta}^{(p)}(u)$ with respect to $k$. This conjecture extends the Macmahon-module construction; the source does not state a resolution.