The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\sigma _{i}^{2}}{\Big )}^{{-3/2}}\cdot \sum \limits _{{i=1}}^{n}\rho _{i}$ and $\sigma_i,\rho_i$ are second and third moments respectively. $\widehat{F_n(x)}$ is the cumulative cdf for $\frac{X_1+\cdots+X_n}{\sqrt{\sigma^2_1+\cdots \sigma^2_n}}$

The DKW bound stated that $$\Pr {\Bigl (}\sup _{{x\in {\mathbb{R}}}}|F_{n}(x)-F(x)|>\varepsilon {\Bigr )}\leq 2e^{{-2n\varepsilon ^{2}}}\qquad {\text{for every }}\varepsilon >0$$.

**Question.** Is it possible to construct such an example $F(x)$ such that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\sim O(\psi_0(n))$$
and simultaneously
$$\Pr {\Bigl (}\sup _{{x\in {\mathbb{R}}}}|F_{n}(x)-F(x)|>\varepsilon {\Bigr )}\sim O(e^{-2n\epsilon})$$
as $n\rightarrow \infty$.

(Note: My motivation is to find such a distribution worse enough to attain these two bounds tightly asymptotically, not to prove "$F=\Phi$" as @Linden pointed out, this may be a possible confusion of motivation.)