= Solution
{c}
Write the unknown common eigenbasis as an unknown unit <Bloch vector> $\mathbf n$. Then $\rho_A$ has vector $0.8\mathbf n$, while $\rho_B$ has vector $-0.4\mathbf n$. A single deterministic channel working for every possible basis must therefore map $0.8\mathbf n$ to $-0.4\mathbf n$ for every unit $\mathbf n$.
By the <basis-independent conversion of qubit Bloch radii>, linearity forces that channel to be $\mathcal E_{-1/2}$. Explicitly, add the conditions for $\mathbf n$ and $-\mathbf n$ to get $\mathcal E(I)=I$, then use the three coordinate directions to fix its action on every Pauli matrix. But $-1/2<-1/3$, so part <3/c/iv> excludes complete positivity. Thus \b[the forward conversion cannot be implemented reliably without knowing the basis].
The reverse conversion would require $\mathcal E_{-2}$, which is not even positive by part <3/c/ii>. Therefore \b[the reverse conversion is also impossible]. Unitary operations interspersed with physically allowed depolarizing maps cannot evade these restrictions: their composition remains a physical <quantum channel>. The question concerns one basis-independent operation on the supplied qubit, not an additional resource of many copies for learning its basis.
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