Zeros of the combination of the Eisenstein series for \(\Gamma _0^+(2)\)

For even integers \(k\ge \ell \ge 4\) $k\ge \ell \ge 4$ , we consider the modular forms \(E_k^+E_\ell ^++E_{k+\ell }^+\) $E_k^{+}{E}_{\ell }^{+}+E_{k+\ell }^{+}$ for the Fricke group \(\Gamma _0^+(2)\) ${\mathrm{\Gamma }}_0^{+}(2)$ , where \(E_k^+\) $E_k^{+}$ is the Eisenstein series of weight k for \(\Gamma _0^+(2)\) ${\mathrm{\Gamma }}_0^{+}(2)$ , and we prove that if \(26630\le \ell < k \le 77\ell \) $26630\le \ell <k\le 77\ell$ or \(k=\ell \ge 10\) $k=\ell \ge 10$ , then all of their zeros in the fundamental domain  \(\mathfrak {F}^+\) ${\mathfrak{F}}^{+}$ for \(\Gamma _0^+(2)\) ${\mathrm{\Gamma }}_0^{+}(2)$ lie on the arc boundary of \(\mathfrak {F}^+\) ${\mathfrak{F}}^{+}$ .