Odd Artin–Schreier symplectic trace conjecture

Assume $p>2$, $d$ is odd, and $(d,p)=1$. Let $\mathcal{O}_d$ be the subfamily of odd polynomials in $\mathcal{F}_d$, and let $T_{f,\psi}^r$ be the $r$-th power sum of the normalized zeroes of $L_{f,\psi}$. Define

$$e_{2,r}=\begin{cases}0,&2\nmid r,\\\\1,&2\mid r.\end{cases}$$

Odd Artin–Schreier symplectic trace conjecture. There exists a constant $\delta>0$ such that, for every $\varepsilon>0$,

$$\left\langle T_{f,\psi}^r\right\rangle_{f\in\mathcal{O}_d}=\begin{cases}-e_{2,r},&r<d,\\\\0,&r\ge d\end{cases}+O_{\varepsilon}\left(q^{\varepsilon r-\delta d}+q^{-\delta r}\right).$$

This is the trace-moment prediction from the unitary symplectic random-matrix model for the odd family; the paper states it as a conjecture because only a much weaker result is proved.