 Solution

Implement the quantum channel by a unitary dilation with a pure initial environment. The final state on $RBE$ is pure. The Schmidt decomposition across either bipartition gives equal complementary Von Neumann entropies, so $S(RB)=S(E)$ and $S(B)=S(RE)$. By the Subadditivity of Von Neumann entropy,The reference is untouched and purifies the input, hence $S(R)=S(\rho )$. Therefore $\boxed{I_c(\Phi ,\rho )\le S(\rho )}$.