Remove polylogarithmic dimension factors in high-dimensional Berry--Esseen bounds for $m$-dependent sums

Let $q>2$, $m,n,d\in\mathbb N$, and let $X_1,\dots,X_n$ be random vectors in $\mathbb R^d$. Assume:

1. $X_i$ are mean-zero: $\mathbb E[X_i]=0$ for all $i$.

2. $X_1,\dots,X_n$ are $m$-dependent, meaning that for every $k\in\{1,\dots,n\}$, the sigma-fields $\sigma(X_i:i\le k)$ and $\sigma(X_j:j\ge k+m+1)$ are independent.

3. A uniform $q$-moment bound holds: for some finite constant $M_q$, $\max_{1\le i\le n}\max_{1\le j\le d}\mathbb E|X_{ij}|^q\le M_q^q$.

4. For

$\Sigma_n$ is nondegenerate in coordinates, e.g. $\min_{1\le j\le d}(\Sigma_n)_{jj}\ge \underline{\sigma}^2$ for some $\underline{\sigma}>0$.

Let $Z\sim N(0,\Sigma_n)$, and let $\mathcal H_d$ be the class of axis-aligned hyper-rectangles

Define the rectangle Kolmogorov distance