  Coulomb branch chiral rings of $\mathcal N=2$ SCFTs are conjectured to be
freely generated. While no counter-example is known, no direct evidence for the
conjecture is known either. We initiate a systematic study of SCFTs with
Coulomb branch chiral rings satisfying non-trivial relations, restricting our
analysis to rank 1. The main result of our study is that (rank-1) SCFTs with
non-freely generated CB chiral rings when deformed by relevant deformations,
always flow to theories with non-freely generated CB rings. This implies that
if they exist, they must thus form a distinct subset under RG flows. We also
find many interesting characteristic properties that these putative theories
satisfy which may behelpful in proving or disproving their existence using
other methods.
