Lee–Zhang conjecture on length-spectrum domination by Fuchsian representations

Let $S$ be a connected closed oriented surface with fundamental group $amma$, and let $L_\rho$ denote the length spectrum of a representation with respect to the specified invariant Finsler metric on the symmetric space $\operatorname{PSL}(n,\mathbb{R})/\operatorname{PSO}(n)$. A representation $\rho:\Gamma\to\operatorname{PSL}(n,\mathbb{R})$ is Hitchin, and $j$ is Fuchsian, as in the source.

Lee–Zhang conjecture. For any Hitchin representation $\rho:\Gamma\to\operatorname{PSL}(n,\mathbb{R})$, there is a Fuchsian representation $j$ such that

$$L_j\leq L_\rho.$$

The conjecture would imply a sharper collar lemma for Hitchin representations. It is proved in the source when $n=3$, but Labourie explained that it cannot hold for $n\geq4$, so the conjecture is refuted in general.