The generalized Mahler conjecture for rational bases

Let $p>q\geq 1$ be coprime integers with $p\leq q^2$. A positive real number $x$ is a $Z_{p/q}$-number if the sequence of fractional parts

$$(\\{x(p/q)^n\\})_{n\in\mathbb N}$$

is contained in $[0,1/q)$. The generalized Mahler conjecture. No $Z_{p/q}$-number exists. This generalizes Mahler's conjecture for the base $3/2$ and is presented as a consequence of the rational-base normality conjecture, but remains unproved in the supplied text.