The non-decoupling conjecture for Hitchin and maximal Higgs bundles

Let $[(E,\phi)]$ be a Higgs bundle in the Hitchin section or one corresponding to a maximal $\operatorname{Sp}(2n,\mathbb R)$-representation, and let $H$ solve the Hitchin equation for $(E,\phi)$.

Non-decoupling conjecture. The Hitchin equation never decouples:

$$F_{\nabla_{\bar\partial_E,H}}\neq 0,\qquad [\phi,\phi_H^*]\neq 0.$$

The source says this is slightly stronger than the negative curvature conjecture and that it is known in several special cases, including $\operatorname{PSL}(2,\mathbb R)$, certain $\operatorname{Sp}(4,\mathbb R)$ components, and cyclic Higgs bundles. It remains open in general.