Let $X$, $Y$, $Z$ be discrete random variables, with $Y$ and $Z$ independent. Does the following equality hold if $Z$ is independent also of $Y$? $$ \max_{f_{Y,Z}} \big\{ \ I(X; f_{Y,Z}(Y,Z)) \ \big\} = \max_{f_X, f_Y} \big \{ \ I(X; f_Y(Y), f_Z(Z)) \ \big \} $$ where the maximization is taken over all non-injective, deterministic functions.

P.S.: See this for the inequality version of the question.