Lower bound conjecture for the top Lyapunov exponent on Hitchin components

Let $C$ be a compact Riemann surface, and let the $n$-th Hitchin component be the connected component of

$$\operatorname{Hom}(\pi_1(C),\operatorname{SL}_n(\mathbb{R}))//\operatorname{SL}_n(\mathbb{R})$$

containing symmetric powers of Fuchsian representations. For a representation in this component, let the top Lyapunov exponent denote its largest Lyapunov exponent. Lower bound conjecture. The top Lyapunov exponent function on the $n$-th Hitchin component is unbounded and greater than or equal to $n-1$, which is the top Lyapunov exponent of the $(n-1)$-th symmetric power of the uniformizing representation. This conjecture proposes an analogue for Lyapunov exponents of the known upper bound phenomenon for critical exponents on Hitchin components; the authors' experiments in rank three suggest both unboundedness and the lower bound, but a proof is not known.