Hitchin-section maximality conjecture in a Hitchin fiber

Let $(\tilde{E},\tilde{\phi})$ be a Higgs bundle in the Hitchin component, and let $(E,\phi)$ be a distinct polystable $SL(n,\mathbb{C})$-Higgs bundle in the same Hitchin fiber. Let $f$ and $\tilde f$ be the corresponding harmonic maps, with energy densities $e(f)$ and $e(\tilde f)$, and let $g_{G/K}$ denote the metric on the symmetric space target. Hitchin-section maximality conjecture. The harmonic maps satisfy

$$e(f)<e(\tilde f),\qquad f^*g_{G/K}<\tilde f^*g_{G/K},$$

and consequently

$$E(f)<E(\tilde f).$$

The conjecture proposes that the Hitchin-section point is maximal for energy density, pullback metric, and total energy within each Hitchin fiber. It is presented as an expected strengthening of the preceding flow question, and no resolution is supplied in the source.