The q-Virasoro cohomology conjecture

Let $Q_n^{+[t]}$ be the maps in the complex associated with the screening charge $Q_n^+$, and let ${\cal H}_{J;n',n}$ denote the proposed irreducible highest weight representation space of the $q$-Virasoro algebra with central charge $c$ and highest weight $h_{J;n',n}=h_{J,J;n',n}$. The q-Virasoro cohomology conjecture. The complex has nontrivial cohomology only at $t=0$:

$$\operatorname{Ker} Q_n^{+[t]}/\operatorname{Im} Q_n^{+[t-1]}=\begin{cases}0,&t\ne0,\\{\cal H}_{J;n',n},&t=0.\end{cases}$$

The conjecture identifies the zero-degree cohomology with the irreducible $q$-Virasoro representation, while the source provides no evidence that the conjecture has been resolved.