Hitchin–Goldman unique energy-minimum conjecture

Let $S$ be a compact surface, let $\mathcal T$ be its Teichmüller space, and let $\rho$ be a representation in Hitchin's component. For each complex structure $j\in\mathcal T$, let $e(j)$ denote the energy of the corresponding harmonic mapping.

Hitchin–Goldman conjecture. The function $e$ has a unique minimum.

The conjecture asks whether, for every representation in Hitchin's component, there is a unique complex structure for which the associated harmonic mapping is minimal. It is known for $n=2$ and $n=3$; for general $n$, the source notes that $e$ is proper, but does not establish uniqueness.