On the semiproportional character conjecture

Two characters of a finite group $G$ are semiproportional if they are not proportional and $G$ is a union of two disjoint normal subsets such that the restrictions of these characters to each of the subsets are proportional. We obtain some results on the structure of an arbitrary finite group having a pair of semiproportional irreducible characters; in particular, assertions on the order of the group and on the kernels of semiproportional characters. We also consider the following conjecture: Semiproportional irreducible characters of a finite group have equal degrees. We validate this conjecture for 2-decomposable groups and prove that if the conjecture holds for two groups then it holds for their direct product.