Kazama–Suzuki duality conjecture for subregular and principal W-algebras

Let $n$ be the parameter in the families of vertex algebras, and set

$$k_1=-n+4,\qquad k_2=-n+1.$$

Here $f_{\rm pr}$ and $f_{\rm sub}$ denote respectively principal and subregular nilpotent elements. Kazama–Suzuki duality conjecture. There is a Kazama–Suzuki type duality between

$$\mathcal W_{k_1}(\mathfrak{sl}(1\vert n),f_{\rm pr})\quad\text{and}\quad \mathcal W_{k_2}(\mathfrak{sl}_{n+2},f_{\rm sub}).$$

This generalizes the explicitly established relation between the corresponding coset algebras and is motivated by the known Kazama–Suzuki duality for related subregular and principal W-algebras. The conjecture asserts that the necessary parameter relation is sufficient in this family; its general validity remains open in the source.