  Optimized quantum $f$-divergence was first introduced by Wilde in
\cite{Wil18}. Wilde raised the question of whether the monotonicity of
optimized quantum $f$-divergence can be generalized to maps that are not
quantum channels. We answer this question by generalizing the monotonicity of
optimized quantum $f$-divergences to positive trace preserving maps satisfying
a Schwarz inequality.
