Almost-optimal bulk regularity conditions in the CLT for G$\beta$E

We consider linear statistics ${\sum }_{k=1}^{N}f(x_k)$ of a Gaussian $\text{b}$-ensemble. Following the work of \cite{LaSo22}, which was concerned only with Wigner matrices, we extend their results for general $\text{b}$. In particular, we show that the CLT holds for any $f\in H^{1\mathrm{/}2+\text{e}}$. The novel input which was not used in \cite{LaSo22} is the Helffer-Sjöstrand representation for the covariance, first appearing the random matrix context in \cite{ErYa15}, coupled with characteristic flow estimates for DBM, first appearing in \cite{Bo22}.