Find the maximum value of
$$\int_{0}^{1} \int_{0}^{1} \left| f(x) - f^{-1}(y) \right|^{3} \,\mathrm d x \,\mathrm d y$$
over weakly decreasing functions $f:[0,1] \to [0,1]$, where $$f^{-1}(y) := \inf\left\{x : f(x) \le y \right\}$$
I am trying to find out how one should proceed with such a problem. I will welcome any partial solution, ideas or good tips. I even wonder if solving this problem is fairly easy or terribly complicated task.