Li's pointwise monotonicity conjecture along the Higgs-bundle flow

Let $[(E,\phi)]$ be a Higgs bundle in $\mathcal M_{\rm Higgs}(\operatorname{SL}(n,\mathbb C))$, let $t\cdot[(E,\phi)]$ be its $\mathbb C^*$-flow, and let $f_{t\cdot[(E,\phi)]}$ be the corresponding equivariant harmonic map. Denote its energy density by $e(f_{t\cdot[(E,\phi)]})$.

Li's monotonicity conjecture. Along the $\mathbb C^*$-flow, the energy density decreases pointwise as $|t|$ decreases.

The integrated energy is known to decrease as $|t|$ decreases, while the source reports the pointwise statement for stable cyclic Higgs bundles. The general case remains open in the source.