There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the generating currents of the resulting W-algebra and their OPE coefficients.

My question is whether a similar algorithm has been worked out for the case where one starts with not simply an affine Lie algebra, but a larger W-algebra that contains the affine Lie algebra as a subalgebra.

An example of this would be to perform the reduction directly on a free field realization of an affine Lie algebra.