Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the Jacobian of $\phi$ and $\phi^{-1}$ are bounded everywhere by some constant $c$, and that $\phi$ is in a Hölder class $C^\alpha(\mathbb{R}^n)$ for some $\alpha\geq 1$ (the space of $\lfloor \alpha\rfloor$ times continuously differentiable functions, with $\lfloor \alpha\rfloor$th derivative $(\alpha-\lfloor \alpha\rfloor)-$Hölder continuous), with $\|\phi\|_{C^\alpha}\leq L$.

Is the following assertion true? If $\alpha\geq |s|$, then there exists a constant $C=C(p,n,s,\alpha,c,L)$ such that, for $f\in H^s_p(\mathbb{R}^n)$ (one can further assume that $f$ is supported on the unit ball if necessary) $$\|f\circ \phi\|_{H^s_p} \leq C\|f\|_{H^s_p}.$$

If $s$ is an integer, then straightforward computations show that the proposition holds, but for $s\not\in \mathbb{N}$, I could not find any reference. The textbook "Theory of function spaces" by Triebel and the article "Mappings of Homogeneous Groups and Imbeddings of Functional Spaces" by Vodopyanov (among others) study this type of question, but only for $\alpha$ an integer.