A counterexample to an optimistic guess about \'etale local systems

Abstract

In p-adic Hodge theory, it is known that if a Galois representation is de Rham, then it becomes semistable after extension of the base field. Liu and Zhu asked whether a corresponding result holds in the relative setting: given an \'etale local system on a quasi-compact rigid analytic variety (for example, a projective scheme) over a p-adic field, does it become semistable after a finite extension of the base field? We give a counterexample.