Rational-coefficient representation conjecture for the modular anomaly function

Let $\mathscr{T}_n^\ell$ be the set of labelled trees with $n$ vertices, let $E_\mathcal{T}$ be the edge set of a tree $\mathcal{T}$, and let $\Gamma_e$ denote the quantity associated with an edge $e$. For a subset $\mathcal{J}\subseteq E_\mathcal{T}$, write $\mathcal{T}_\mathcal{J}$ for the corresponding tree data, and let $\delta_{\Gamma_e}$ and $\operatorname{sgn}(\Gamma_e)$ denote the delta and sign factors appearing below. Then

$$\mathscr{E}^{(0)}_n(\{\check\gamma_i\}) = \frac{1}{n!} \sum_{\mathcal{T}\in\, \mathds{T}_n^\ell} \kappa_\mathcal{T}(\{\check\gamma_{\mathfrak{v}}\})\, S_\mathcal{T}(\{\check\gamma_\mathfrak{v}\}),$$

where

$$S_\mathcal{T}(\{\check\gamma_\mathfrak{v}\})= \sum_{\mathcal{J}\subseteq E_\mathcal{T}}e_{\mathcal{T}_\mathcal{J}} \,\prod_{e\in \mathcal{J}}\delta_{\Gamma_e} \prod_{e\in E_\mathcal{T}\setminus \mathcal{J}} \operatorname{sgn}(\Gamma_e).$$

Rational-coefficient representation conjecture. The coefficients $e_{\mathcal{T}}$ are rational numbers depending only on the topology of $\mathcal{T}$. This proposed representation is intended to replace the formulation whose tree coefficients can be irrational, consistently with the expectation that the completion of a mock modular form with rational Fourier coefficients contains only rational numbers.