Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is positive. Consider a pants decomposition containing $x$. Consider the Fenchel-Nielsen coordinate of the Teichmuller space with respect to this pants decomposition. Let $l_{\chi_s}(y)$ denote the length function of $y$ with respect to the left hand side twist parameter $s$, associated to the the curve $x$. It seems that there exist a $s_0$ such that $l_{\chi_s}(b)$ is an increasing function for $s>s_0$. Is this true? If yes, what is the proof?

Thanks in advance.

PS 1: I know that the function $l_{\chi_s}(b)$ is strictly convex.