Galois invariance implies gradedness for associated Higgs bundles

Let $\mathfrak X$ be the model appearing in the paper, and let $(E,\theta)$ be a Higgs bundle over $\mathfrak X_{{\mathcal O}_{\widehat{\overline k}}}$. Suppose the isomorphism class of its associated generalized representation is fixed by the action of $\operatorname{Gal}(\bar k/k)$.

Gradedness conjecture. Then $(E,\theta)$ is graded.

The source presents this statement after a rank-$2$ proposition and gives an argument for the rank-$2$ case. It is intended as the corresponding general statement for Higgs bundles whose associated generalized representations are Galois-invariant.