Kedlaya's prediction on bounded-support p-adic algebraic numbers

Let the support of a $p$-adic Hahn expansion be the ordered set of exponents with nonzero coefficients, and let its order type be the order type of that set. Kedlaya's prediction. There does not exist any $p$-adic algebraic number with bounded support of order type $\omega$. Kedlaya observed that only finite orders, $\omega$, and $\omega^\omega$ might occur, while the paper notes that no $p$-adic algebraic number with bounded support of infinite order type is currently known. The conjecture remains open.