Central limit conjecture for mean-field interacting boson random point fields

Let $u_{\kappa,\beta,\mu}$ be the distribution of the random point measure $4\xi_{\kappa}4$ on $4\mathbb{R}^d4$ for $4\kappa>04$, and let $4\mathfrak{L}4$ denote Lebesgue measure on $4\mathbb{R}^d4$. Write $4G(\beta)4$ for the operator appearing in the covariance expression. Central limit conjecture. As $4\kappa\to\infty4$, the random field

$$\frac{\pi^{d/4}}{\kappa^{d/4}}\Big(\frac{\lambda}{\mu-\mu_{\lambda,c}(\beta)}\Big)^{1/2}\Big(\xi_{\kappa}-\frac{\kappa^{d/2}}{\pi^{d/2}}\frac{\mu-\mu_{\lambda,c}(\beta)}{\lambda}\mathfrak{L}\Big)$$

converges in distribution to the Gaussian random field on $4\mathbb{R}^d4$ with covariance

$$(1+G(\beta))(1-G(\beta))^{-1}.$$

This is suggested by the small-test-function expansion of the generating functional and expresses Gaussian fluctuations around the macroscopic mean density in the condensed regime; the paper presents it as a conjectural central limit theorem rather than proving it.