  Fix an affine Lie algebra $\widehat{\mathfrak{g}}_\kappa$ with associated
principal affine W-algebra $\mathcal{W}_\kappa$. A basic conjecture of
Frenkel--Kac--Wakimoto asserts that Drinfeld--Sokolov reduction sends
admissible $\widehat{\mathfrak{g}}_\kappa$-modules to zero or cohomological
shifts of minimal series $\mathcal{W}_\kappa$-modules. We prove this conjecture
and a natural generalization to the spectrally flowed Drinfeld--Sokolov
reduction functors. This extends the previous results of Arakawa for the minus
reduction.
