Monotonicity conjecture for energy density along the Higgs-bundle b1-action

Let $\mathcal{M}_{Higgs}$ be the moduli space of $SL(n,\mathbb{C})$-Higgs bundles, with the action

$$t\cdot(E,\phi)=(E,t\phi)$$

of $\mathbb{C}^*$. For a corresponding harmonic map, write $e(f)$ for its energy density. Monotonicity conjecture. Along the $\mathbb{C}^*$-flow, the energy density of the corresponding harmonic maps is monotonically increasing as $|t|$ increases.

Hitchin proved monotonicity of the total energy along this flow, while the conjecture asks for the stronger pointwise monotonicity of energy density. The conjecture is known for cyclic Higgs bundles by work of Dai and the author; the general case remains open.