The braid-reversed vertex-tensor-category conjecture for D^ψ(n,m)

Let $m,n\geq2$, let $\psi$ be generic, and set $k=\psi-m+1$ and $\ell=\phi-n+1$, where $\phi^{-1}+\psi^{-1}=1$. Let $\mathcal H$ be a rank-one Heisenberg algebra. The D-algebra category conjecture. The vertex tensor category of modules for $\mathcal D^\psi(n,m)\otimes\mathcal H$ should be braid-reversed equivalent to the Deligne product

$$KL_k(\mathfrak{sl}_m)\boxtimes KL_\ell(\mathfrak{sl}_n).$$

This conjecture supplies the categorical gluing data expected to produce the relevant vertex-algebra extensions. The source gives no resolution status.