The Fuchsian length domination conjecture for Hitchin representations

Let $S$ be the surface and $\rho$ a representation in $Hit_n(S)$. A representation $\rho'$ is in the Fuchsian locus of $Hit_n(S)$ when it is a Fuchsian representation viewed in $Hit_n(S)$, and $l_\rho(\gamma)$ denotes the length associated with $\rho$ for $\gamma\in\Gamma$. Fuchsian length domination conjecture. There is some representation $\rho'$ in the Fuchsian locus of $Hit_n(S)$ such that

$$l_{\rho}(\gamma) \geq l_{\rho'}(\gamma), \qquad \text{for any } \gamma \in \Gamma.$$

This conjecture would compare the marked length spectrum of every Hitchin representation with that of a Fuchsian representation. The supplied passage says that the authors were unable to show that the classical collar inequality fails in $Hit_n(S)$ for any $n>2$, motivating this conjecture; no resolution is given here.