Given a function $f: mathbb R to mathbb R$, we say $g: mathbb R to mathbb R$ is an topological rescaling of $f$ if $g = fh$ for some orientation preserving homeomorphism $h$ of $mathbb R$.

Given a sequence $f_n: mathbb R to mathbb R$ of absolutely continuous functions, do there exist topological rescalings $g_n$ of $f_n$ (where the choice of $h$ is allowed to depend on $n$) such that the restricted functions ${g_n}_{lvert(0, 1)}$ are equicontinuous?