Connectivity conjecture for fixed-point components of real Higgs-bundle moduli spaces

Let $G$ be one of $U(p,p)$, $SU(p,p)$, or $Sp(2p,2p)$, and let $\Theta_{G}$ be the involution defined in the source. Consider the Hitchin fibration

$$\mathcal{M}_{G^{c}}\rightarrow \mathcal{A}_{G^{c}}.$$

Connectivity conjecture. Every connected component of the fixed-point set of $\Theta_{G}$ intersects the generic fibre of this Hitchin fibration.

This conjecture relates the connectivity of moduli spaces of real $G$-Higgs bundles to the geometry of the Hitchin fibration. The preceding results establish related connectivity statements for components intersecting regular fibres, but the conjecture asserts the corresponding property for every fixed-point component and the generic fibre.