In quantum source coding, we have an encoder $\mathcal{E}$ and a decoder $\mathcal{D}$ which are some quantum channels. Given a state $\rho_A$ on Hilbert space $\mathcal{H}_A$, we wish to encode and then decode it to obtain
$$(\mathcal{D}\circ\mathcal{E})(\rho_A) = \rho_A'$$
and we want that such a $\rho_A'$ is close to the original state $\rho_A$. One can use the fidelity as a measure of success to see how well the source coding was done i.e. we want a large
$$F(\rho_A, \rho_A').$$
Definition 5.3 of https://arxiv.org/pdf/1403.2543.pdf looks at a different criterion. For any purification $\psi_{AR}$ of $\rho_A$, it looks at the entanglement fidelity
$$F_e(\psi_{AR}, \left((\mathcal{D}\circ\mathcal{E})\otimes I_R\right)\psi_{AR})$$
By Uhlmann's theorem, it must hold that $$F(\rho_A, \rho_A') \geq F_e(\psi_{AR}, \left((\mathcal{D}\circ\mathcal{E})\otimes I_R\right)\psi_{AR})$$
Is the reverse inequality true? Can the entanglement fidelity criterion be lower bounded using the fidelity criterion somehow in this case?
