The semi-infinite-cohomology simple-superalgebra conjecture

Let $m,n$ and the levels $k,\ell$ be as in the source, and let $\pi^{k-\ell}$ be the rank-one Heisenberg vertex algebra. Semi-infinite-cohomology conjecture. The relative semi-infinite Lie algebra cohomology

$$H^{\mathrm{rel},0}_{\infty}(\mathfrak{sl}_m,\mathcal W^\psi(n-m,m)\otimes A[\mathfrak{sl}_m,1-\psi]\otimes\pi^{k-\ell})$$

should be a simple vertex superalgebra. The construction is intended to produce odd generators of conformal weight $\frac{n+1}{2}$ and relate triality-related algebras through semi-infinite cohomology. The source gives no resolution status.