A freely generated ring for $N=1$ models in class $S_k$

We study 4d $N=1$ supersymmetric theories of class $S_k$, obtained from flux compactifications on a Riemann surface of 6d $(1,0)$ conformal theories describing the low energy physics on a stack of M5 branes probing a $Z_k$ singularity. We conjecture that the protected spectrum of class $S_k$ theories contains a freely generated ring, generalizing the Coulomb branch of the $N=2$ theories. We derive this by examining a limit of the supersymmetric index of 4d $N=1$ class $S_k$ theories. The limit generalizes the Coulomb limit of $N=2$ theories, which coincides with the case of $k=1$ for a particular choice of flux. We conjecture a general simple formula for the index in the aforementioned limit.