Universal $2$-parameter $\mathcal{N}=2$ supersymmetric $\mathcal{W}_{\infty}$-algebra

The universal $2$-parameter vertex algebra $\mathcal{W}_{\infty}$ of type $\mathcal{W}(2,3,\dots)$ is a classifying object for vertex algebras of type $\mathcal{W}(2,3,\dots,N)$ for some $N$; under mild hypotheses, all such vertex algebras arise as quotients of $\mathcal{W}_{\infty}$. In 2017, Gaiotto and Rap\v{c}\'ak introduced a family of such vertex algebras called $Y$-algebras, and conjectured that they fall into groups of three that are mutually isomorphic. This is a common generalization of both Feigin-Frenkel duality and the coset realization of principal $\mathcal{W}$-algebras in type $A$, and was proven in 2021 for the simple $Y$-algebras (i.e., one label is zero) by the first and third authors. In this paper, we extend this entire story to the $\mathcal{N}=2$ superconformal setting. First, we prove the 2013 conjecture of Gaberdiel and Candu that there exists a universal $2$-parameter vertex algebra $\mathcal{W}^{\mathcal{N}=2}_{\infty}$ which is an extension of the $\mathcal{N}=2$ superconformal algebra, and has four additional generators in weights $i, i + \frac{1}{2}, i + \frac{1}{2}, i+1$, for each integer $i > 1$. This admits many $1$-parameter quotients which we call $\mathcal{N}=2$ supersymmetric $Y$-algebras, and we prove the dualities among these algebras which were conjectured in 2018 by Prochazka and Rap\v{c}\'ak. A special case is the coset realization of the principal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{sl}_{n+1|n})$ which was conjectured in 1992 by Ito. As a corollary, we obtain the strong rationality of $\mathcal{W}_k(\mathfrak{sl}_{n+1|n})$ for $k = -1 + \frac{1}{n+a+1}$ for all positive integers $n,a$, and we describe its module category. This generalizes Adamovi\'c's 1999 result on $\mathcal{N}=2$ minimal models, which is the case $n=1$.