Logarithmic level-rank duality for boundary VOAs

Let $\widetilde{{\mathcal N}}_{n,k}$ be the slight modification of the boundary VOA ${\mathcal N}_{n,k}$ obtained by successive extension and orbifold, let ${\mathcal D}_{n,k}\simeq\mathcal{FT}_k(\mathfrak{sl}_n)$, and let ${\text{FF}}(nk)$ be $nk$ copies of free fermions. Logarithmic level-rank duality conjecture. The two VOAs are mutual commutants inside the free fermions:

$$\widetilde{{\mathcal N}}_{n,k}\simeq {\text{FF}}(nk)/{\mathcal D}_{n,k},\qquad {\mathcal D}_{n,k}\simeq {\text{FF}}(nk)/\widetilde{{\mathcal N}}_{n,k}.$$

This would induce a braid-reversed equivalence of their module categories and, consequently, an equivalence of corresponding derived categories. It is presented as a novel logarithmic level-rank duality and remains open.