Is a quasiconvex function always the monotone image of a convex function?

Question

Not every quasiconvex function is convex. However, given a quasiconvex function $f:X\to \Bbb{R}$ (where $X$ is a convex subset of $\Bbb{R}^n$), can we always find a monotone or at least non-decreasing $m:\Bbb{R}\to\Bbb{R}$ and a convex function $\tilde f:X\to\Bbb{R}$ such that $m\circ \tilde f= f$ ?