Geodesic-coordinate conjecture for the pressure metric on the Hitchin component

Let $S$ be a closed oriented surface with genus $g\geq 2$ and $n\geq 4$. For any point $\sigma\in \mathcal{T}(S) \subset \mathcal{H}_{n}(S)$, let $X$ be the Riemann surface corresponding to $\sigma$. The Hitchin parametrization

$$\bigoplus\limits_{i=2}^{i=n} H^{0}(X,K^i)$$

provides geodesic coordinates for the pressure metric at $\sigma$. The conjecture extends the theorem established in the paper for the previously treated case to all Hitchin components $\mathcal{H}_{n}(S)$ with $n\geq 4$; whether this geodesic-coordinate property holds in those higher-rank cases remains open.