Properly colored $C_{4}$'s in edge-colored graphs

When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results forcing rainbow triangles to appear, we consider similar conditions which force the existence of a properly colored copy of $C_{4}$.
Comment: corrected a statement after Theorem 4