Complete contraction and expansion conjecture for amplitude damping channels

Let $\mathcal A_{\gamma}$ denote the amplitude damping channel with damping parameter $\gamma$, and for two channels define the complete contraction and expansion coefficients by

$$\eta_{\mathcal N,\mathcal M}^{cb}=\sup_{\rho_{VA}}\frac{I(V;B_1)}{I(V;B_2)},\qquad \widecheck{\eta}_{\mathcal N,\mathcal M}^{cb}=\inf_{\rho_{VA}}\frac{I(V;B_1)}{I(V;B_2)},$$

where the optimization ranges over all valid states $\rho_{VA}$ on $\mathcal H_V\otimes\mathcal H_A$. Complete contraction and expansion conjecture. If $0<\gamma_2<\gamma_1<1$, then

$$\eta_{\mathcal A_{\gamma_1},\mathcal A_{\gamma_2}}^{cb}<1,\qquad \widecheck{\eta}_{\mathcal A_{\gamma_1},\mathcal A_{\gamma_2}}^{cb}>0.$$

This is proposed as a sufficient condition for positivity of the ratio $R_3$ for the pair of amplitude damping channels; its resolution is not established in the supplied text.