5 problems

Let $X$ be a projective smooth curve over $\mathbf C=\mathbf C_p$, the complete algebraic closure of $\mathbf Q_p$. A semistable Higgs bundle on $X$ is a Higgs bundle of degree $0$…

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

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 $k$…

Descending conjecture. The geometric sublocal system in Theorem 1.2 descends to a ${\mathbb Z}_p$-étale subrepresentation

Faltings' conjecture. The Higgs bundle $(E,\theta)$ corresponds to a ${\mathbb C}_p$-geometric representation.