An undecidability result for the asymptotic theory of $p$-adic fields

Fix a prime $p$. We prove that the set of sentences true in all but finitely many finite extensions of ${\mathbb{Q}}_p$ is undecidable in the language of valued fields with a cross-section. The proof goes via reduction to characteristic $p$, adapting Pheidas’ proof of the undecidability of ${\mathbb{F}}_p(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$ with a predicate for powers of $t$. This answers a variant of a question of Derakhshan-Macintyre.