Is minimizing trace distance between two density matrices equivalent to minimizing their entropy difference?

Suppose $\rho$ is an input state , and $\sigma$ is a recovered state obtained after approximately encoding $\rho$ in a quantum error-correcting code and then applying some recovery channel. Here, the minimization is over the choice of recovery channel . We fix the encoding map $\mathcal{E}$ and the input state $\rho$ . Let $\mathcal{R}$ be any CPTP map acting on the code space. The recovered state is $$ \sigma(\mathcal{R}) = \mathcal{R} \circ \mathcal{E}(\rho). $$ The optimizations we are compar