Minimum trace-distance contraction conjecture for qubit channels over a fixed classical channel

Let $a,f\in[0,1]$ and let $\mathcal{Q}_{\mathbb{C}}(a,f)$ denote the set of qubit channels over the classical channel

$$\begin{pmatrix} 1-a & a\\\\ 1-f & f \end{pmatrix}.$$

For a qubit channel $Q$, let $\eta^{\operatorname{Tr}}(Q)$ be its trace-distance contraction coefficient. Minimum trace-distance contraction conjecture.

\inf\left\\{\eta^{\operatorname{Tr}}(Q):Q\in\mathcal{Q}_{\mathbb{C}}(a,f)\right\\}=|a-f|.

This value is the trace-distance contraction coefficient of the underlying classical channel. The paper establishes the corresponding supremum but leaves this infimum conjectural.