On zeros of Eisenstein series for genus zero Fuchsian groups

Let $\Gamma\leq$SL$_{2}(\mathbb{R})$ be a genus zero Fuchsian group of the first kind with $\infty$ as a cusp, and let $E_{2k}^{\Gamma}$ be the holomorphic Eisenstein series of weight $2k$ on $\Gamma$ that is nonvanishing at $\infty$ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on $\Gamma,$ and on a choice of a fundamental domain $\mathcal{F}$, we prove that all but possibly $c(\Gamma,\mathcal{F})$ of the nontrivial zeros of $E_{2k}^{\Gamma}$ lie on a certain subset of $\{z\in\mathfrak{H}\,:\,j_{\Gamma}(z)\in\mathbb{R}\}$. Here $c(\Gamma,\mathcal{F})$ is a constant that does not depend on the weight, $\mathfrak{H}$ is the upper half-plane, and $j_{\Gamma}$ is the canonical hauptmodul for $\Gamma.$