The biquadratic-curve one-level density conjecture

Let $q$ be odd, and let $\mathcal{H}_{g,\operatorname{biq}}$ denote the moduli space of biquadratic curves of genus $g$ over $\mathbb{F}_q$. For a curve $C$ in this space, let $\Theta_C$ be its Frobenius class, and let $f$ be an even Schwartz function with Fourier transform $\hat{f}$ supported in $(-\alpha,\alpha)$. Write $Z_f(\Theta_C)$ for the associated one-level density statistic, and define

$$USp(2g)^3=\left\{\begin{pmatrix}U_1&0&0\\0&U_2&0\\0&0&U_3\end{pmatrix}:U_1,U_2,U_3\in USp(2g)\right\}.$$

The biquadratic-curve one-level density conjecture. As $g\to\infty$,

$$\left\langle Z_f(\Theta_C)\right\rangle_{\mathcal{H}_{g,\operatorname{biq}}}=\int_{USp(2g)^3}Z_f(U)\,dU+o(1).$$

This conjecture predicts that the one-level density for biquadratic curves has the symplectic symmetry type represented by three independent copies of $USp(2g)$. The paper proves the corresponding trace-moment statement only in a sublinear range, so the asserted one-level density remains open.