Random projection of norm rate of invariance

Let $X$ be a random vector in $\mathbb{R}^n$ with independent and identically distributed entries,

$$X=\begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.$$

Assume that $\mathbb{E}x_1=0$, $\mathbb{E}x_1^2=1$, and $\mathbb{E}x_1^4=1+\sigma^2$ for $0<\sigma<\infty$. Let $S:\mathbb{R}^n\to\mathbb{R}^m$ be a random matrix with independent entries satisfying $\mathbb{E}S_{i,j}=0$ and $\mathbb{E}S_{i,j}^2=1$. Assume that $S$ and $X$ are independent and that $\mathbb{E}S_{1,1}^8\vee\mathbb{E}|x_1|^6<c<\infty$. Let $G$ be a standard normal random variable. Random projection of norm rate of invariance. Then

$$\sup_t\left|\mathbb{P}\left(\frac{X^TS^TSX-mn}{\sqrt{\sigma^2m^2n+2mn^2}}<t\right)-\mathbb{P}(G<t)\right|\le O\left(\frac{1}{\sqrt n}+\frac{1}{\sqrt m}\right),$$

and

$$\sup_t\left|\mathbb{P}\left(\frac{X^TS^TSX-mn}{\sqrt{\sigma^2m^2n+2mn^2}}<t\right)-\mathbb{P}\left(\frac{X^TX-n}{\sigma\sqrt n}<t\right)\right|\le O\left(\frac{1}{\sqrt n}+\frac{1}{\sqrt m}\right).$$

These bounds quantify the rate at which the distribution of the randomly projected norm approaches a Gaussian law and the corresponding standardized norm of $X$; the source presents them as a conjecture based on detailed calculations, and does not provide resolution evidence.