  We describe an $N=2$ supersymmetric Poisson vertex algebra structure of $N=1$
(resp. $N=0$) classical $W$-algebra associated with $\mathfrak{sl}(n+1|n)$ and
the odd (resp. even) principal nilpotent element. This $N=2$ supersymmetric
structure is connected to the principal $\mathfrak{sl}(2|1)$-embedding in
$\mathfrak{sl}(n+1|n)$ superalgebras, which are the only basic Lie
superalgebras that admit such a principal embedding.
