Conjecture on Poissonian correlations for prime rotations

Let $\alpha$ be a badly approximable number, meaning that there is a constant $c>0$ such that

$$\|q\alpha\|\geq \frac{c}{q}$$

for every positive integer $q$. Consider the prime rotation sequence $(p_n\alpha)_{n \in \mathbb{N}} \mod 1$, where $p_n$ denotes the $n$-th prime.

Prime-rotation conjecture. The prime rotation sequence $(p_n\alpha)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders and, consequently, has Poissonian gap distribution.

The conjecture proposes Poissonian fine-scale statistics for prime rotations at every badly approximable rotation parameter. The source presents it as an expected extension of the paper's result beyond the assumption that the auxiliary function tends to zero; no resolution is given here.