Gaussian number-fluctuation conjecture for large intervals

Let $E_\beta^{(\cdot)}(n;J)$ denote the probability of exactly $n$ particles in $J$, let $n_J$ be the corresponding particle number, and let $\langle n_J\rangle$ and $\operatorname{Var}n_J$ denote its mean and variance. Gaussian number-fluctuation conjecture. For $n\approx\langle n_J\rangle$,

$$E_\beta^{(\cdot)}(n;J)\mathop{\sim}\limits_{|J|\to\infty}\frac{1}{\sqrt{2\pi\operatorname{Var}n_J}}\exp\left(-\frac{(n-\langle n_J\rangle)^2}{2\operatorname{Var}n_J}\right).$$

For bulk, soft and hard scaling, respectively, the source also gives the corresponding leading asymptotics of the means and variances. This is a macroscopic central-limit prediction for the number of particles in a large interval.