Conjectural asymptotics for the moment generating function of linear statistics

Fix $\beta>0$. Let $g:\mathbb{R}\to\mathbb{R}$ be $2\pi$-periodic and belong to $C^{1,q}$ for some $0<q\leq 1$, let $t\in\mathbb{R}$, and let $I$ denote the integral whose normalized ratio gives the moment generating function of the linear statistic. Moment-generating-function asymptotics conjecture. As $n\to\infty$,

$$I\left(\frac{t}{\sqrt n}g\right)=2^{\beta n(n-1)/2}\left(\frac{8\pi}{\beta n}\right)^{(n-1)/2}\sqrt n\,e^{-1/(2\beta)}(1+o(1))\int_{-\pi}^{\pi}\exp\left(t\sqrt n\,g(\theta)+\frac{2g'(\theta)^2}{\beta}t^2\right)\,d\theta,$$

uniformly for $t$ in compact subsets of $\mathbb{R}$. In particular,

$$\mathbb{E}\left[e^{\frac{t}{\sqrt n}\sum_{j=1}^n g(\theta_j)}\right]=\frac{I(\frac{t}{\sqrt n}g)}{I(0)}=\frac{1+o(1)}{2\pi}\int_{-\pi}^{\pi}\exp\left(t\sqrt n\,g(\theta)+\frac{2g'(\theta)^2}{\beta}t^2\right)\,d\theta.$$

These asymptotics would provide the moment-generating-function basis for the conjectured subleading fluctuation law. The statement is presented after the authors explain that the required regime is substantially more involved and that existing methods do not directly apply, so it remains open.