There is the following exercise in Vershynin's book on High-Dimensional Probability.

Exercise 3.1.6: Let $X = (X_1, \dots, X_n) \in \mathbb{R}^n$ be a random vector with independent coordinates $X_i$ that satisfy $\mathbb{E}[X_i^2] = 1$ and $\mathbb{E}[X_i^4] \leq K^4$. Show that $\mathrm{Var}(\|X\|_2) \leq CK^4$.

What if we remove the condition that the coordinates are independent? Suppose instead that we have only the condition that the coordinates are uncorrelated (i.e., $X$ has covariance $I$), and we additionally have some bound on the fourth moment (e.g., $\mathbb{E} [ \langle X, v\rangle^4]\leq K^4$). What is the best bound we can prove on $\mathrm{Var}(\|X\|_2)$?

There is the following exercise in Vershynin's book on High-Dimensional Probability.

Exercise 3.1.6: Let $X = (X_1, \dots, X_n) \in \mathbb{R}^n$ be a random vector with independent coordinates $X_i$ that satisfy $\mathbb{E}[X_i^2] = 1$ and $\mathbb{E}[X_i^4] \leq K^4$. Show that $\mathrm{Var}(\|X\|_2) \leq CK^4$.

What if we remove the condition that the coordinates are independent? Suppose instead that we have only the condition that the coordinates are uncorrelated (i.e., $X$ has covariance $I$), and we additionally have some bound on the fourth moment (e.g., $\mathbb{E} [ \langle X, v\rangle^4]\leq K^4$ for every unit vector $v$). What is the best bound we can prove on $\mathrm{Var}(\|X\|_2)$?