Invariance of semi-infinite cohomology under the higher-genus ab-move

Let $\mathbf{V}_{G_u,3}$ be the trinion vertex algebra with maximal untwisted punctures and let $\mathbf{V}_{1,1}$ be the mixed trinion vertex algebra. Let $i_1,i_2,i_3$ denote the three actions of $V^{\kappa_c}(\mathfrak{g}_u)$ on $\mathbf{V}_{G_u,3}$, and let $j_2,j_3$ denote the actions of $V^{\kappa_c}(\mathfrak{g}_t)$ on $\mathbf{V}_{1,1}$. The affine algebras act through the indicated pairs of puncture actions: $\hat{\mathfrak{g}}_{u,-\kappa_g}$ acts on $\mathbf{V}_{G_u,3}$ via $i_2\otimes(i_3\circ\sigma)$, where $\sigma$ is the $\mathbb{Z}_2$ outer automorphism, while $\hat{\mathfrak{g}}_{t,-\kappa_g}$ acts on $\mathbf{V}_{1,1}$ via $j_2\otimes j_3$. Semi-infinite cohomology ab-move conjecture. The semi-infinite cohomologies are isomorphic:

$$\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{u,-\kappa_g},\mathfrak{g}_u,\mathbf{V}_{G_u,3})\cong \mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{t,-\kappa_g},\mathfrak{g}_t,\mathbf{V}_{1,1})\,.$$

This conjecture would establish invariance under the $ab$-move, completing the generalized $S$-duality relations beyond the moves already controlled by associativity and the $4$-moves. The paper states that a proof of this invariance is not known; its status is therefore open.