Berry-Esseen Theorem-like result with fourth central moment instead of third absolute moment

Berry-Esseen Theorem-like result with fourth central moment instead of
third absolute moment

Let $X_i$, $i=1,\ldots,n$ be i.i.d. random variables with $E[X_i]=\mu$,
$E[(X_i-\mu)^2]=\sigma^2$, and $E[(X_i-\mu)^4]=\kappa$. I am interested in
approximating the distribution of
$Y_n=\frac{X_1+\ldots+X_n}{\sqrt{n}\sigma}$ by the standard normal
distribution $\mathcal{N}(0,1)$. By Berry-Esseen Theorem we know that the
total variation distance between the c.d.f. $F_n(x)$ of $Y_n$ and the
c.d.f. $\Phi(x)$ of $\mathcal{N}(0,1)$ is upper-bounded as follows:
$$\tag{1}|F_n(x)-\Phi(x)|\leq \frac{C\rho}{\sigma^3\sqrt{n}}$$
where $\rho=E[|X_i-\mu|^3]$ is the absolute third moment of $X_i$ and $C$
is a constant.
In my particular problem, the fourth central moment $E[(X_i-\mu)^4]$ is
much easier to compute than the third absolute moment $E[|X_i-\mu|^3]$.
While the $E[(X_i-\mu)^4]$ provides a ready upper bound for
$E[|X_i-\mu|^3]$ per answers to my previous question, which I can just
plug into (1), obtaining same asymptotics, I am wondering if there is a
tighter result than that, specific to the fourth central moment.