Guichard–Wienhard conjecture for extra components of $SO(p+q,q)$-Higgs-bundle moduli

Let $\tau$ denote extension data in the spectral-data description of an $SO(p+q,q)$-Higgs bundle, and write its spectral data as $(L,M,\tau)$. Let $\bar{\pi}$ be the relevant spectral covering map, let $I$ be a line bundle on the Riemann surface $\Sigma$, and let $\mathcal{O}^{\oplus q}$ denote the trivial rank-$q$ bundle.

Guichard–Wienhard conjecture. The extra components in the moduli space of $SO(p+q,q)$-Higgs bundles conjectured to contain positive representations are those connected components containing Higgs bundles whose spectral data has the form

$$(L,M,\tau)=(\bar{\pi}^*I,\mathcal{O}^{\oplus q},\tau),$$

where $I$ is a non-trivial $\mathbb{Z}_2$-line bundle on the Riemann surface $\Sigma$.

This identifies a proposed family of extra connected components using their spectral data; the source attributes the conjecture to Guichard and Wienhard, while its resolution is not specified here.