Galois invariance and descent of graded Higgs bundles

Let $k$ be the base field and let a generalized representation be associated to a Higgs bundle. A generalized representation is Galois-invariant when its isomorphism class is fixed by the action of $\operatorname{Gal}(\bar k/k)$, and a Higgs bundle is graded when its Higgs field has the grading form used in the paper.

Galois-invariance conjecture. A generalized representation is $\operatorname{Gal}(\bar k/k)$-invariant if and only if the corresponding Higgs bundle is graded and defined over $k$ up to an isomorphism.

The paper proves this equivalence for rank-$2$ generalized representations and states that it is expected to hold in every rank. Thus the unresolved aspect is the extension from rank $2$ to arbitrary rank.