Let $U$ be a simply connected open bounded subset of $\mathbb{C}\cong\mathbb{R}^{2}$ with smooth boundary (thus the boundary $\partial U$ is diffeomorphic to a circle). Let $L\geq0$ denote the total arc of $\partial U$, and let $\phi\colon[0,L)\to\partial U$ be an arc length parametrization of $\partial U$.
If $0\leq s\leq t\leq L$, then is the arc length of the curve $$\{\phi(x): s\leq x\leq t\}$$ is given by $t-s$, and therefore it is geometrically clear that $|\phi(t)-\phi(s)|\leq t-s$.
Question: Does there exist a constant $C\geq0$ such that $|t-s|\leq C|\phi(t)-\phi(s)|$ for all $s,t\in[0,L)$?
In other words, is the inverse $\phi^{-1}\colon\partial U\to[0,L)$ Lipschitz?
Since $\phi$ is an arc length parametrization it satisfies $|\phi'(x)|=1$ for all $0\leq x<L$.
The function $\phi^{-1}\colon\partial U\to[0,L)$ not even continuous: For $z_n = \phi(1/n)$ and $w_n = \phi(L-1/n)$ is $$ w_n - z_n = \phi(L-1/n)-\phi(1/n) \to 0 $$ but $$ \phi^{-1}(w_n)-\phi^{-1}(z_n) = L-2/n \to L \, . $$