Anticoncentration of Complex Gaussian Hafnians

Let $G_{2n}$ be a complex symmetric random matrix whose entries above the diagonal are independent standard circular complex Gaussians, and let $H_n=\operatorname{haf}(G_{2n})$. We prove the uniform shifted anticoncentration bound $$ \Pr\!\left( \left| \frac{H_n}{\sqrt{(2n-1)!!}}-z \right| \le \vare