Rudnick–Sarnak Poissonian local statistics conjecture for quadratic sequences

Let $Y$ be uniformly distributed on $[0,1)$, and for $N\in\mathbb{N}$ and $0<L\leq N$ define the random variable

$$W_{\alpha,L,N}:=\left\lvert\{n\leq N:\alpha n^2\in[Y,Y+L/N]\ \operatorname{mod}1\}\right\rvert.$$

Here $\operatorname{Po}(L)$ denotes a Poisson-distributed random variable with parameter $L$. Rudnick–Sarnak conjecture. For almost all $\alpha\in[0,1]$, for every fixed $L>0$,

$$W_{\alpha,L,N}\xrightarrow{\operatorname{dist}}\operatorname{Po}(L)$$

as $N\to\infty$. This is a strong local form of pseudorandomness for the sequence $(\alpha n^2\ \operatorname{mod}1)_{n=1}^{\infty}$ and is equivalent to Poissonian spacing statistics. The source presents it as a conjecture and gives no resolution evidence.