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