On the determinantal range of matrix products

Let matrices $A,C\in M_n$ have eigenvalues $\alpha_1,\dots,\alpha_n$ and $\gamma_1,\dots,\gamma_n$, respectively. The set $D_C(A)=\{\det(A-UCU^*)\colon U\in M_n,\ U^*U=I_n\}$ of complex numbers is called the $C$-determinantal range of $A$. The paper studies various conditions under which the relation $D_C(RS)=D_C(SR)$ holds for some matrices $R$ and $S$.