On zeros of Eisenstein series for genus zero Fuchsian groups

Let $\text{GN}\le \text{SLR}$ be a genus zero Fuchsian group of the first kind with $\mathrm{\infty }$ as a cusp, and let $\text{Ek}$ be the holomorphic Eisenstein series of weight $2k$ on $\text{GN}$ that is nonvanishing at $\mathrm{\infty }$ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on $\text{GN},$ and on a choice of a fundamental domain $\text{F}$, we prove that all but possibly $c(\text{GN},\text{F})$ of the non-trivial zeros of $\text{Ek}$ lie on a certain subset of $\{z\in \mathfrak{H}:\text{JN}(z)\in \mathbb{R}\}$. Here $c(\text{GN},\text{F})$ is a constant that does not depend on the weight $2k$ and $\text{JN}$ is the canonical hauptmodul for $\text{GN}\mathrm{.}$