Completely contractive data-processing conjecture for trace-norm conditional mutual information

Let $\mathcal{E}:C\rightarrow C'$ be a quantum channel. Define its local trace-norm contraction ratio on $C$ by

$$\eta_{1,C}:=\sup_{\rho_C,\rho'_C}\frac{\lVert\mathcal{E}_C[\rho_C]-\mathcal{E}_C[\rho'_C]\rVert_1}{\lVert\rho_C-\rho'_C\rVert_1}.$$

For a tripartite state $\rho_{ABC}$, let $I_1(A:C|B)_\rho$ denote the trace-norm conditional mutual information.

Completely contractive data-processing conjecture for trace-norm conditional mutual information. If

$$\eta_{1,C}<1,$$

then there exists a global constant $\eta<1$ such that for any tripartite system $ABC$ and any state $\rho_{ABC}$,

$$I_1(A:C'|B)_{\mathcal{E}(\rho)}\leq\eta I_1(A:C|B)_\rho.$$

This is proposed as a trace-norm analogue of the conditional-mutual-information contraction conjecture. The supplied context does not state whether this formulation is open or known, so its status remains open.