Bounded-support conjecture for p-adic algebraic Hahn series

Let $f\in\mathbf{L}_p$ be a $p$-adic algebraic number, and let $\operatorname{Supp}(f)$ denote its support. Suppose that $\operatorname{Supp}(f)$ is bounded.

Bounded-support conjecture. The support $\operatorname{Supp}(f)$ must be a finite set.

This conjecture was formulated as a consequence of Kedlaya's prediction that only finite order types, $\omega$, and $\omega^\omega$ can occur as order types of supports of $\mathbf{Q}_p$-algebraic $p$-adic Hahn series. It was previously shown to be implied by that prediction; its resolution is not specified here.