The nilpotent-cone domination conjecture for Higgs-bundle energy density

Let $[(E,\phi)]$ be a Higgs bundle in $\mathcal M_{\rm Higgs}(\operatorname{SL}(n,\mathbb C))$, let $t\cdot[(E,\phi)]$ denote its $\mathbb C^*$-flow, and let $f_{[(E,\phi)]}$ be the corresponding equivariant harmonic map. The limit as $t\to0$ lies in the nilpotent cone.

Nilpotent-cone domination conjecture. The energy density satisfies

$$e(f_{[(E,\phi)]})\geq e\left(f_{\lim_{t\to0}t\cdot[(E,\phi)]}\right).$$

The source states that this is proved for every Higgs bundle in the Hitchin section, but remains open in general.