Larcher–Stockinger conjecture on non-Poissonian pair correlations

Let $(a_n)_{n \in \mathbb{N}}$ be an integer sequence. For a real number $\alpha$, consider the sequence $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ and its pair correlations.

Larcher–Stockinger conjecture. If for almost all $\alpha$ the pair correlations of $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ are not Poissonian, then the pair correlations of this sequence are not Poissonian for any $\alpha$.

The conjecture concerns whether failure of Poissonian pair correlations for almost every rotation parameter must hold for every parameter. It is refuted in the paper: the sequence $(a_n^{(f)})_{n \in \mathbb{N}}$, with $f$ as in the cited theorem, provides a counterexample.