A Lyapunov-type bound in $R^d$

Let $\fs X1n$ be independent random vectors taking values in $R^d$ such that ${E X_k =0}$ for all $k$. Write ${S=\fsu X1n}$. Assume that the covariance operator, say $C^2$, of $S$ is invertible. Let $Z$ be a centered Gaussian random vector such that covariances of $S$ and $Z$ are equal. Let $\mathscr{C}$ stand for the class of all convex subsets of $R^d$. We prove a Lyapunov-type bound for $\Delta =\sup_{A\in\mathscr{C}}|P\{S\in A\}-P\{Z\in A\}|$. Namely, ${\Delta \le c d^{1/4} \beta}$ with ${\beta =\fsu \beta 1n}$ and ${\beta_k= E |C^{-1}X_k|^3}$, where $c$ is an absolute constant. If the random variables ${\fs X1n}$ are independent and identically distributed and $X_k$ has identity covariance, then the bound specifies to ${\Delta \le c d^{1/4} E |X_1|^3/\sqrt{n}}$. Whether one can remove the factor $d^{1/4}$ or replace it with a better one (eventually by $1$), remains an open question.