Conjecture on Gaussian fluctuations of zero–critical-point distances

For each zero $Z_{k,n}$ of a random polynomial $p_n$, let $\zeta_{k,n}$ be a nearest critical point, and define

$$d_{k,n}:=|Z_{k,n}|^2\sqrt{\frac n{\log n}}\left(n(Z_{k,n}-\zeta_{k,n})-\frac1{\overline{Z}_{k,n}}\right),$$ $$\chi_n:=\sum_{k=1}^n\delta_{d_{k,n}}.$$

Here $\mathcal N_{\mathbb C}(0,1)$ denotes the standard normal distribution on $\mathbb C$, and convergence is convergence in probability of the associated random measures. Gaussian fluctuation conjecture. For random polynomials with i.i.d. zeroes,

$$\chi_n\overset{P}{\longrightarrow}\mathcal N_{\mathbb C}(0,1),\qquad\text{as }n\to\infty.$$

This is the refined fluctuation prediction for the pairing between zeroes and critical points, motivated by the second-order approximation in the source and supported there by numerical simulations; no proof or resolution is given in the supplied material.