Coherent Information and Entanglement Breaking channels

The book by John Watrous, "The Theory of Quantum Information" is an exciting read for anyone wanting to research quantum information theory. The following question presumes some background covered in the book, which I will do my best to explain.

Question:
If $mathcal{X}_0, mathcal{X}_1, mathcal{Y}_0, mathcal{Y}_1$ are complex Euclidean spaces (Finite dimensional Hilbert spaces on complex field) and $Phi_0 in C(mathcal{X}_0,mathcal{Y}_0)$ $Phi_1 in C(mathcal{X}_1,mathcal{Y}_1)$ are two quantum channels, with $Phi_0$ being entanglement breaking, then show that
begin{equation}
I_C(Phi_0otimesPhi_1) = I_C(Phi_1).
end{equation}

Background:
Here, the space of channels $C(mathcal{X}_0, mathcal{Y}_0)$ consists of linear, completely positive trace-preserving (CPTP) maps on $L(mathcal{X}_0)$ (linear operators on $mathcal{X}_0$) returning operators in $L(mathcal{Y}_0)$. For any channel $Phi$ and state $S$ (states are non-negative operators with unit trace), the coherent information $I_C(Phi;S) = H(Phi(S)) – H((Phiotimes I)(vec(sqrt{S})vec(sqrt{S})^dagger))$ and $I_C(Phi) = max_S I_C(Phi;S)$, where the maximum is taken with respect to all states. For any operator $A = sum_{i,j}a_{ij}|e_iranglelangle f_j|$, $vec(A) = sum_{i,j}a_{ij}|e_irangle otimes |f_jrangle$.

A separable state is a state of the form $sum_i p_i P_iotimes Q_i$, where $P_i$ and $Q_i$ are states with $sum_i p_i = 1$ with $p_ige 0$. States that do not have this kind of representation are called entangled. An entanglement breaking channel is one whose output state is always separable even if the input is entangled.

There is a theorem that may prove useful: The following are equivalent.

1. $Phi$ is entanglement breaking (EB).
2. For any finite-dimensional Hilbert space $mathcal{Z}$, $Phiotimes I_mathcal{Z}$ is separable where $I_mathcal{Z}$ is the identity map in $mathcal{Z}$.
3. There exist states $rho_i$ and non-negative operators $M_i$ such that $sum_{i}M_i = I$ (called a POVM) such that $Phi(S) = sum_i rho_i Tr(SM_i)$.

I was able to show $I_C(Phi_0 otimesPhi_1) ge I_C(Phi_0) + I_C(Phi_1)$ for any two channels $Phi_0$ and $Phi_1$. The book does prove that $I_C(Phi) le 0$ for $Phi$ EB. But I’ve been unable to make any progress beyond that.

I will be happy to offer any clarification if required. This problem has been bugging me for quite some time now and I cannot see a way through. This problem would interest students and professors working in Quantum Information Theory.

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Cross-posted on math.SE