Equality of lower and upper accumulation exponents for Pisot numbers

Let $\zeta$ be a Pisot number and let $\alpha$ be real. The quantities $\underline{\sigma}(\alpha,\zeta)$ and $\overline{\sigma}(\alpha,\zeta)$ denote the lower and upper rates of accumulation of $\alpha\zeta^n$ modulo $1$ to $0$. Equality conjecture. For every such $\alpha$ and $\zeta$, one has

$$\underline{\sigma}(\alpha,\zeta)=\overline{\sigma}(\alpha,\zeta).$$

The preceding results establish bounds for these exponents, including $\underline{\sigma}(\alpha,\zeta)=0$ when $\alpha$ is real and outside $\mathbb{Q}(\zeta)$. The conjecture asserts that the lower and upper rates nevertheless always coincide, and the paper gives no proof of this equality.