The descending conjecture for geometric subrepresentations

Let ${\mathbb L}$ be an étale local system and let ${\mathbb W}^\text{geo}_{\mathcal O_{{\mathbb C}_p}}$ be the geometric sublocal system produced in Theorem 1.2. A subrepresentation is ${\mathbb Z}_p$-étale when it is defined over ${\mathbb Z}_p$ as an étale representation.

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

$${\mathbb W}\subset {\mathbb L}.$$

Consequently, ${\mathbb W}$ is a de Rham representation.

The conjecture asks whether the geometric subrepresentation can first descend to an ${\mathcal O}_{{\mathbb C}_p}$-subrepresentation and then have its coefficient ring reduced to ${\mathbb Z}_p$. The paper proposes an approach to the first step using a $p$-adic analogue of Simpson's ${\mathbb C}^*$-action.