Dai–Li's Hitchin-fiber maximality conjecture

Let $\mathcal M_{\rm Higgs}(\operatorname{SL}(n,\mathbb C))$ be the Higgs-bundle moduli space, and let a Hitchin fiber be a fiber of the Hitchin fibration. Let $[(\widetilde E,\widetilde\phi)]$ lie in the Hitchin section and let $[(E,\phi)]$ be a distinct polystable $\operatorname{SL}(n,\mathbb C)$-Higgs bundle in the same Hitchin fiber. Write $f_{[(E,\phi)]}$ and $f_{[(\widetilde E,\widetilde\phi)]}$ for the corresponding harmonic maps.

Dai–Li's conjecture. The energy densities satisfy

$$e(f_{[(E,\phi)]})<e(f_{[(\widetilde E,\widetilde\phi)]}),$$

and hence

$$f_{[(E,\phi)]}^*g_N<f_{[(\widetilde E,\widetilde\phi)]}^*g_N.$$

Consequently, the energies satisfy

$$E(f_{[(E,\phi)]})<E(f_{[(\widetilde E,\widetilde\phi)]}).$$

Equivalently, inside each Hitchin fiber the maximum of the energy occurs exactly at the image of the Hitchin section. The source reports the $n=2$ case as proved, while for $n\geq3$ even the integrated version remains open.