The folklore conjecture on Hardy pairs

Let

$$H:= \{ (\lambda, \alpha) \in \mathbb{R}^{\times} \times (1, \infty): \|\lambda \alpha^n\| \to 0 \text{ as } n \to \infty \}$$

be the Hardy set, where $\|x\|$ denotes the distance from $x$ to the nearest integer. The folklore conjecture. If $(\lambda, \alpha) \in H$, then $\alpha$ is a Pisot–Vijayaraghavan number and $\lambda$ lies in $\mathbb{Q}(\alpha)$.

This conjecture proposes a complete answer to Hardy's question in the setting of the Hardy set. Hardy's theorem establishes the assertion when $\alpha$ is assumed to be algebraic; the general case remains open.