  In this work we study the rate of convergence in the central limit theorem
for the Euclidean norm of random orthogonal projections of vectors chosen at
random from an $\ell_p^n$-ball which has been obtained in [Alonso-Guti\'errez,
Prochno, Th\"ale: Gaussian fluctuations for high-dimensional random projections
of $\ell_p^n$-balls, Bernoulli 25(4A), 2019, 3139--3174]. More precisely, for
any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension
$k_n\in\{1,\ldots,n\}$, $P_{E_n}$ the orthogonal projection onto $E_n$, and
$X_n$ be a random point in the unit ball of $\ell_p^n$. We prove a Berry-Esseen
theorem for $\|P_{E_n}X_n\|_2$ under the condition that $k_n\to\infty$. This
answers in the affirmative a conjecture of Alonso-Guti\'errez, Prochno, and
Th\"ale who obtained a rate of convergence under the additional condition that
$k_n/n^{2/3}\to\infty$ as $n\to\infty$. In addition, we study the Gaussian
fluctuations and Berry-Esseen bounds in a $3$-fold randomized setting where the
dimension of the Grassmannian is also chosen randomly. Comparing deterministic
and randomized subspace dimensions leads to a quite interesting observation
regarding the central limit behavior. In this work we also discuss the rate of
convergence in the central limit theorem of [Kabluchko, Prochno, Th\"ale:
High-dimensional limit theorems for random vectors in $\ell_p^n$-balls, Commun.
Contemp. Math. (2019)] for general $\ell_q$-norms of non-projected vectors
chosen at random in an $\ell_p^n$-ball.
