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

P.S.: See [this][1] for the inequality version of the question.


  [1]: https://mathoverflow.net/questions/426330/maximization-of-information-over-set-of-non-injective-functions