  We examine a strong/weak duality between a Heisenberg coset of a theory with
$\mathfrak{sl}_n$ subregular $\mathcal{W}$-algebra symmetry and a theory with a
$\mathfrak{sl}_{n|1}$-structure. In a previous work, two of the current authors
provided a path integral derivation of correlator correspondences for a series
of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper,
we derive correlator correspondences in a similar way but for a different
series of generalized duality. This work is a part of the project to realize
the duality of corner vertex operator algebras proposed by Gaiotto and
Rap\v{c}\'ak and partly proven by Linshaw and one of us in terms of two
dimensional conformal field theory. We also examine another type of duality
involving an additional pair of fermions, which is a natural generalization of
the fermionic FZZ-duality. The generalization should be important since a
principal $\mathcal{W}$-superalgebra appears as its symmetry and the properties
of the superalgebra are less understood than bosonic counterparts.
