Pairwise modular inequivalence of zeros in the half-strip

Prove that, for every integer n>0, all zeros of the multiple Eisenstein quasimodular form G_{\{2\}^n} in the half-strip are pairwise SL_2(\mathbb{Z})-inequivalent.

Background

The paper proves that every zero of G_{{2}n} is simple and that each such form has infinitely many SL_2(\mathbb{Z})-inequivalent zeros in the half-strip. It does not, however, establish that no two zeros in that half-strip belong to the same SL_2(\mathbb{Z}) orbit.

The authors note that the assertion follows from the transformation law of G_2 when n=1, but remains unproved for general n, including n=2. The conjecture asks for the stronger classification that every zero in the half-strip represents a distinct modular orbit.

References

The following conjecture follows immediately from the transformation law of $G_2$ when $n=1$ (See Proposition 3.3). However, for general $n$, it remains unproved even in the case $n=2$.

\begin{conj} For each integer $n>0$, all zeros of $G_{{2}n}$ in the half-strip are pairwise $SL_2(Z)$-inequivalent in the half-strip. \end{conj}

— Zeros of Quasimodular Forms Defined by Iterated Sums  (2609.12729 - Kina et al., 11 Sep 2026) in Section 1, subsection “Summary of our results”