Undecidability of expansions of Laurent series fields by cyclic discrete subgroups

In 1987, Pheidas showed that the field of Laurent series ${\mathbb{F}}_q((t))$ with a constant for the indeterminate $t$ and a predicate for the natural powers $\{t^n\mid n>0\}$ of $t$ is existentially undecidable. We show that the same result holds true if $t$ is replaced by any element $\alpha$ of positive $t$-adic valuation.