Rudnick–Sarnak conjecture on Poissonian gap statistics for polynomial sequences

Let $\varphi(z)=\alpha_q z^q+\alpha_{q-1}z^{q-1}+\cdots+\alpha_0$ be a polynomial of degree $q$, and let $P_N(\,\cdot\,;\varphi)$ denote the gap distribution of $\{\varphi(j)\}_{j=1}^N$ modulo $1$, with gaps normalized by $N$. Rudnick–Sarnak conjecture. Let $q\geq 2$. There is a set $Q\subset\mathbb{R}$ of full Lebesgue measure such that, for $\alpha_q\in Q$ and any interval $A\subset[0,\infty)$,

$$\lim_{N\to\infty}P_N(A;\varphi)=\int_A e^{-s}\,\operatorname{d}s.$$

The claim predicts exponential, equivalently Poissonian, limiting gap statistics for almost every leading coefficient of a polynomial of degree at least two. It is presented as a conjecture of Rudnick and Sarnak; the supplied text gives no resolution status.