Ando-type unitarily invariant norm inequalities for operator functions

Let $A$ and $B$ be positive matrices, let $f$ be a function on $[0,\infty)$ with $f(0)=0$, and let $|||\cdot|||$ be an arbitrary unitarily invariant norm. Ando-type conjecture. If $f$ is operator monotone, then

$$||| (A-B)(f(A)-f(B))||| \le ||||A-B|f(|A-B|)|||.$$

If $f$ is operator convex, the inequality is reversed. This would extend the Ando inequality from the operator norm setting to arbitrary unitarily invariant norms for operator monotone and operator convex functions.