  We prove the longstanding physics conjecture that there exists a unique
two-parameter $\mathcal{W}_{\infty}$-algebra which is freely generated of type
$\mathcal{W}(2,3,\dots)$, and generated by the weights $2$ and $3$ fields.
Subject to some mild constraints, all vertex algebras of type
$\mathcal{W}(2,3,\dots, N)$ for some $N$ can be obtained as quotients of this
universal algebra. As an application, we show that for $n\geq 3$, the structure
constants for the principal $\mathcal{W}$-algebras
$\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ are rational
functions of $k$ and $n$, and we classify all coincidences among the simple
quotients $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ for
$n\geq 2$. We also obtain many new coincidences between
$\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ and other vertex
algebras of type $\mathcal{W}(2,3,\dots, N)$ which arise as cosets of affine
vertex algebras or nonprincipal $\mathcal{W}$-algebras
