Let $X_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n \frac{Z_i - \mu}{\sigma}$ be the normalized partial sums of some iid. random variables $Z_1, \dots$ Let $f_n(x)$ be the density function of $X_n$, then the Edgeworth expansion gives us
$$\begin{align} f_n(x) &= \phi(x) \\ &\quad -\frac{1}{n^{\frac{1}{2}}}\left(\tfrac{1}{6}\lambda_3\,\phi^{(3)}(x) \right) \\ &\quad +\frac{1}{n}\left(\tfrac{1}{24}\lambda_4\,\phi^{(4)}(x) + \tfrac{1}{72}\lambda_3^2\,\phi^{(6)}(x) \right)\\ &\quad+O(n^{-3/2}) \end{align}$$ where $\phi(x)$ is the density function of the standard normal distribution and $\lambda_k = \kappa_k/\sigma^k$ where $\kappa_k$ is the $k$th cumulant.
Since this is a true asymptotic expansion we have $|f_n(x) - \phi(x)| \le C \lambda_3 n^{-1/2}$ for some constant, $C$, uniform over $x$.
Unfortunately, $C$ can still depend on the higher cumulants. This is in contrast with the Berry Esseen theorem, which states $$\left|F_n(x) - \Phi(x)\right| \le C \lambda_3 n^{-1/2}$$ (where $F_n$ is the cumulative distribution function for $X_n$ and $\Phi(x)$ for the normal distributino) in which the constant is uniform also over $\lambda$. (E.g. $C=1/2$ suffices).
Question: Is there a way to get make the Edgeworth expansion uniform? Related: Is it possible to expand the Berry Esseen bound to lower order terms, like $an^{-1/2}+Cn^{-1}$ for a precise value $a$ and some bounded constant $C$?