Conjectured centered fluctuation limit for the Cramér–von Mises statistic

Let $H$ be the random matrix and let $\beta$ and the statistic $\mathcal{A}_N$ be as in Theorem. Let $Z_k$, $k\geqslant 1$, denote the random variables appearing in that theorem, and let $a_\beta$ be the constant defined there. Centered fluctuation conjecture. Under the assumptions of Theorem,

$$N^2(\mathcal{A}_N-\mathbb{E}\mathcal{A}_N)\overset{d}{\longrightarrow} \frac{1}{\beta\pi^2}\sum_{k=1}^\infty\left(\frac{1}{k}(Z_k^2-1)-\frac{1}{\sqrt{k(k+2)}}Z_kZ_{k+2}\right)+\frac{2-\beta}{\sqrt{\beta}\pi^2}\sum_{k=1}^\infty\left(\frac{k+2}{4k^{3/2}(k+1)}Z_{2k}-\frac{1}{4k\sqrt{k+1}}Z_{2k+2}\right)+a_\beta.$$

This conjecture is motivated by simulations of the expectation and fluctuation of the Cramér–von Mises statistic; the source does not provide a proof or a resolution, so the conjecture remains open.