Interlacing of zeros of the Eisenstein series for $\varGamma_0^+(2)$

We prove that for $a=2,4$, if $k\geq 16$ is an even integer, then the non-trivial zeros of $E_{k}^+$ and $E_{k+a}^+$ for $\varGamma_0^+(2)$ interlace on the arc of the fundamental domain for the Fricke group $\varGamma_0^+(2)$ of level $2$, where $E_{k}^+(z)$ is the Eisenstein series for $\varGamma_0^+(2)$.