Length-convexity and domination for Hitchin representations

Let $S$ be a closed oriented surface of genus $g\geq 2$. For any $n\geq 3$, let $\text{Hit}_n(S)$ be the Hitchin component and let $\text{Hit}_n(S)^\prime$ be the subset of representations that take values in $\mathrm{PSp}_{2m}(\mathbb{R})$ if $n=2m$ and $\mathrm{PSO}(m,m+1)$ if $n=2m+1$. In this no