  The Bershadsky-Polyakov algebra is the $\mathcal{W}$-algebra associated to
$\mathfrak{s}\mathfrak{l}_3$ with its minimal nilpotent element $f_{\theta}$.
For notational convenience we define $\mathcal{W}^{\ell} = \mathcal{W}^{\ell -
3/2} (\mathfrak{s}\mathfrak{l}_3, f_{\theta})$. The simple quotient of
$\mathcal{W}^{\ell}$ is denoted by $\mathcal{W}_{\ell}$, and for $\ell$ a
positive integer, $\mathcal{W}_{\ell}$ is known to be $C_2$-cofinite and
rational. We prove that for all positive integers $\ell$, $\mathcal{W}_{\ell}$
contains a rank one lattice vertex algebra $V_L$, and that the coset
$\mathcal{C}_{\ell} = \text{Com}(V_L, \mathcal{W}_{\ell})$ is isomorphic to the
principal, rational $\mathcal{W}(\mathfrak{s}\mathfrak{l}_{2\ell})$-algebra at
level $(2\ell +3)/(2\ell +1) -2\ell$. This was conjectured in the physics
literature over 20 years ago. As a byproduct, we construct a new family of
rational, $C_2$-cofinite vertex superalgebras from $\mathcal{W}_{\ell}$
