Semiclassical Moser--Trudinger type inequalities for orthonormal families in $\mathbb{R}^2$

For $H^1(\mathbb{R}^2)$-orthonormal families $\{u_n\}_{n=1}^N$ with one-body density $ρ=\sum_{n=1}^N |u_n|^2$, we prove the whole-space endpoint Moser--Trudinger inequality $$ \int_{\mathbb{R}^2}\frac{e^{4πρ(x)}-1-4πρ(x)}{ρ(x)}\,dx \le C\left(N-1+\int_{\mathbb{R}^2}ρ(x)\,dx\right), $$ where the inte