Undecidability of expansions of Laurent series fields by cyclic discrete subgroups

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