Leading-pole cancellation conjecture for the holomorphic modular ambiguity

Let ${\boldsymbol r}$ be a set of $n$ charges, let $m_{\boldsymbol r}$ be its index, let $E_2(\tau)$ be the second Eisenstein series, and let $\mathcal{A}_0^{\boldsymbol r}$ be the set of residue classes $\boldsymbol\nu$ that allow a solution to the maximal-zero-mode condition. Define

$$c_{\boldsymbol r}=\frac{r_0}{(2^{\epsilon}\pi\mathrm{i}\kappa)^{n-1}r}\prod_{k=1}^{n-1}\left(\sum_{i=1}^{k}r_i\sum_{j=n-k+1}^{n}r_j\right)^{-1}.$$

Leading-pole cancellation conjecture. The choice

$$\phi^{({\boldsymbol r})}_{\boldsymbol\nu}(\tau,z)=\frac{\operatorname{Sym}\{c_{\boldsymbol r}\}}{z^{n-1}}\exp\left(-\frac{\pi^2}{3}m_{\boldsymbol r}E_2(\tau)z^2\right)\delta_{\boldsymbol\nu\in\mathcal{A}_0^{\boldsymbol r}}$$

is conjectured to cancel the leading pole.