Take a Whittaker module $Wh$ of a (finite or affine) semi-simple Lie algebra $\mathfrak{g}$ , and apply the quantum Drinfeld-Sokolov reduction $qDS$ with respect to an $sl(2)$ embedding $\rho:sl(2) \to \mathfrak{g}$.

Is $qDS(Wh)$ a Whittaker module of the corresponding W-algebra $W(\mathfrak{g},\rho)$ ?

If not, how can I characterize $qDS(Wh)$? Another natural question would be, for which module $M$ of $\mathfrak{g}$ is the reduction $qDS(M)$ a Whittaker module of the W-algebra?