Let $\Sigma$ be a surface of finite type. Let $\mathcal{S}$ be the set of non-trivial isotopy classes of simple closed curves on $\Sigma$. One denotes by $l_x(\alpha)$ the infimal length of curves in the class of $\alpha$ in the metric $x$. The metric $x$ can be considered to be a point in the Teichmüller space $\mathcal{T}$ of $\Sigma$ and hence a hyperbolic metric, the length will be realized on a closed geodesic. Thurston introduced the following asymmetric metric on $\mathcal{T}$

$$ L(x, y)=\log \sup _{\alpha \in \mathcal{S}} \frac{l_y(\alpha)}{l_x(\alpha)} $$

We consider a (Hölder) continuous map $\omega \mapsto f(\omega)$ where $f(\omega)$ are homeomorphisms of $\Sigma$ (or more generally semi-contractions of $\mathcal{T}$ ). You can think that $f:\Omega \to \text{Homes}^+ (\Sigma)$, where $\Omega$ is a compact metric space.

Fix a base point $x_0 \in \mathcal{T}$, is is true that there is $0<C<1$ such that for every $\omega$ $$L(x_0, f(\omega) x_0) \geq C?$$

I don't expect that to be true in the above general concept, but I was wondering if it could be true under some minimal conditions (or generic conditions).

I was trying to use this result that said \begin{aligned} &L\left(x_0, y\right)=\log \sup _{\alpha \in \mathcal{S}} \frac{l_y(\alpha)}{l_{x_0}(\alpha)} \asymp \log \max _{\alpha \in \mu} \frac{l_y(\alpha)}{l_{x_0}(\alpha)}\\ &\text { up to an additive error. } \end{aligned}

On the other hand, I thought one could use the definition of the metric; the logarithm of the infimum of Lipschitz constants of any homeomorphism from $x$ to $y$ that is homotopic to the identity.