  We find all values of $k\in \mathbb C$, for which the embedding of the
maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak
g,\theta)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra
and $-\theta$ its minimal root. In particular, it turns out that if
$W_k(\mathfrak g,\theta)$ does not collapse to its affine part, then the
possible values of these $k$ are either $-\frac{2}{3} h^\vee$ or
$-\frac{h^\vee-1}{2}$, where $h^\vee$ is the dual Coxeter number of $\mathfrak
g$ for the normalization $(\theta,\theta)=2$. As an application of our results,
we present a realization of simple affine vertex algebra $V_{-\tfrac{n+1}{2} }
(sl(n+1))$ inside of the tensor product of the vertex algebra
$W_{\tfrac{n-1}{2}} (sl(2| n), \theta)$ (also called the Bershadsky-Knizhnik
algebra) with a lattice vertex algebra.
