We study the 4d, $N=2$ rigid superconformal Yang-Mills theory on a fixed Lorentzian manifold admitting twistor spinors. Whether this classical symmetry can be maintained at the quantum level (i.e. whether one can perturbatively remove the anomaly), is the main concern of this talk. To answer this question, we need to employ an extended version of the BRST formalism, since local gauge and rigid superconformal symmetries are intertwined. The main result is to establish criteria under which the theory can be consistently quantized. More precisely, we prove that the interacting BRST current is conserved and the interacting BRST charge realizes a nilpotent derivation on the enlarged algebra of interacting fields including unphysical and gauge variant fields (ghosts and anti-fields). We then, construct the algebra of physical gauge-invariant renormalized interacting observables as the cohomology of this charge. The result follows from certain Ward identities obtained by prescribing a renormalization scheme, as well as studying certain cohomology classes of the BRST differential. Joint work with Stefan Hollands.