Let $Z\sim\mathcal N_{\mathbb C}(0,1)$ be a complex normal variable, that is $$Z=V+\mathrm iW,$$ where $V,W\sim\mathcal N_{\mathbb R}(0,1/2)$ are independent real normal variables. Consider further a real normal variable $X\sim \mathcal N_{\mathbb R}(0,1)$.
For a bound $\alpha>0$, is it always true that $$\mathbb P(|Z|>\alpha)\leq P(|X|>\alpha)?$$
This is my try to save the conjecture that I proposed in this post (which turned out to be wrong). It turns out that one can (using heuristic arguments) go from the situation in the linked problem to the one presented here.
As in the other question, we can equivalently try to prove $$\mathbb P(V^2+W^2>\alpha)\leq \mathbb P(X^2>\alpha).$$ It is not hard to see that the left-hand side is at least half of the right-hand side, but this of course does not help.
Also, using convolutions seems a little overkill here (I am frankly not even sure if it would lead to anything).
Since you're only looking at the magnitude, the complex part of this drops out completely and you're just left with a $\chi^2$ distribution, $k=1,2$.
This reduces to a mechanical verification that $\chi^2_1$ dominates $\frac{1}{2}\chi^2_2$ in CDF. Wiki gives the CDF as $$F(x;k)=\frac{1}{\Gamma(k/2)}\gamma\left(\frac{k}{2},\frac{x}{2}\right)$$ and in fact the special form $F(x;2)=1-e^{-\frac{x}{2}}$.
A cursory look at the plot of the CDFs show that your conjecture here is wrong. Wolfram plot.