Low-lying zeros conjecture for cubic twists of an elliptic curve over a function field

Let $\boldsymbol{\mathcal{F}_{N}(B)}$ denote the family over which the average is taken, and let $\widetilde{E}$ be an elliptic curve over $\mathbb{F}_q(T)$ given by the minimal Weierstrass equation $y^2=x^3+B$, with $B\formathbb{F}_q[T]$. For $n\in\mathbb{Z}_{\geq1}$, write $\eta_2(n)$ and $\eta_3(n)$ for the divisibility indicators appearing in the formula, and let $\Theta_{\widetilde{E}_D}$, $\Theta_{\widetilde{E}}$, and $\Theta_{\operatorname{sym}^3\widetilde{E}}$ denote the corresponding Frobenius conjugacy classes. Then $\widetilde{\mathcal{D}}(n)$ is a prime sum over degrees dividing $n$ and satisfies the stated bound.

Low-lying zeros conjecture. For every $n\in\mathbb{Z}_{\geq1}$,

$$\left\langle\operatorname{Tr}(\Theta^n_{\widetilde{E}_D})\right\rangle_{\mathcal{F}_{N}(B)}=\eta_2(n)+\frac{\eta_3(n)}{q^{n/3}}\left(\operatorname{Tr}\left(\Theta^{n/3}_{\operatorname{sym}^3\widetilde{E}}\right)+\frac{1}{2}\operatorname{Tr}\left(\Theta^{n/3}_{\widetilde{E}}\right)\right)+\frac{\widetilde{\mathcal{D}}(n)}{q^{n/2}}(1+o(1)).$$

Here $\widetilde{\mathcal{D}}(n)$ is bounded by $q^{n/8}$, although it may be considerably smaller. This is a heuristic conjecture arising from the expected cubic contribution to the one-level density and relating it to the Frobenius of the symmetric-cube $L$-function; the source does not provide a resolution, so the conjecture remains open.