  We propose new strong/weak dualities in two dimensional conformal field
theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality
between Witten's cigar model described by the
$\mathfrak{sl}(2)/\mathfrak{u}(1)$ coset and sine-Liouville theory. In a
previous work, a proof of the FZZ-duality was provided by applying the
reduction method from $\mathfrak{sl}(2)$ Wess-Zumino-Novikov-Witten model to
Liouville field theory and the self-duality of Liouville field theory. In this
paper, we work with the coset model of the type
$\mathfrak{sl}(N+1)/(\mathfrak{sl}(N) \times \mathfrak{u}(1))$ and propose that
the model is dual to a theory with an $\mathfrak{sl}(N+1|N)$ structure. We
derive the duality explicitly for $N=2,3$ by applying recent works on the
reduction method extended for $\mathfrak{sl}(N)$ and the self-duality of Toda
field theory. Our results can be regarded as a conformal field theoretic
derivation of the duality of the Gaiotto-Rap\v{c}\'ak corner vertex operator
algebras $Y_{0, N, N+1}[\psi]$ and $Y_{N, 0, N+1}[\psi^{-1}]$.
