Mesoscopic central limit theorem for general $\beta$-ensembles

We prove that the linear statistics of eigenvalues of $\beta$-log gasses satisfying the one-cut and off-critical assumption with a potential $V \in C^6(\mathbb{R})$ satisfy a central limit theorem at all mesoscopic scales $\alpha \in (0; 1)$. We prove this for compactly supported test functions $f \in C^5(\mathbb{R})$ using loop equations at all orders along with rigidity estimates.