Show $\int_{\gamma}e^{iz}e^{-z^2}dz$ same value on every line parallel to $\mathbb{R}$

From an old qualifier: Show that $$\large\int_{\gamma}e^{iz}e^{-z^2}\mathrm dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. Justify the estimates involved.

My first thought was to draw a long strip and integrate over it. By Cauchy's theorem the integral is zero, and I can compare the contributions. I'd like it if, for well-chosen such strips, the contribution on the sides were entirely imaginary and that on the top/bottom were entirely real, or vice-versa. But so far, no luck.

Take a rectangle with vertices $\pm R, \pm R + iY$, for an arbitrary but fixed $Y$.
On the edges between $R$ and $R+iY$ resp $-R$ and $-R+iY$, the integrand can be estimated
$$\bigl\lvert e^{iz}e^{-z^2} \bigr\rvert \leqslant e^{\lvert Y\rvert}e^{Y^2-R^2},$$
$$\lvert Y\rvert e^{\lvert Y\rvert + Y^2 - R^2} \xrightarrow{R\to +\infty} 0,$$
so since by Cauchy's integral theorem the integral over the rectangle is $0$, the proposition follows.