Approximation of quadratic forms of independent random vectors by accompanying laws

Let $X, X_1,X_2,\dots$ be independent and identically distributed random vectors taking values in $\mathbb{R}^d$. Assume that $\mathsf{E}X=0$, $\mathsf{E}|X|^{8/3}<\infty$ and that $X$ is not concentrated in a proper subspace of $\mathbb{R}^d$. Let $Y,Y_1,Y_2,\dots$ denote i.i.d. random vectors with common distribution which is accompanying to that of $X$. We compare the distributions of the nondegenerate quadratic forms $Q[S_N]$ and $Q[T_N]$ of the normalized sums $S_N=N^{-1/2}(X_1+\dots+X_N)$ and $T_N=N^{-1/2}(Y_1+\dots+Y_N)$ and prove that
$$\begin{align*} &\sup_x|\mathsf{P}\{Q[S_N-a]<x\}-\mathsf{P}\{Q[T_N-a]<x\}| &\qquad=O((1+|a|^4)N^{-1}), \qquad a\in\mathbb{R}^d, \end{align*}$$
provided that $9\le d\le\infty$. The constant in this bound depends on $\mathsf{E}|X|^{8/3}$, $Q$, and the covariance operator of $X$. We also show the optimality of the bound $O(N^{-1})$.