How fast does spectral radius of truncated circular unitary ensemble converge?

Let $z_1,\cdots ,z_p$ be the eigenvalues of $A,$ which is the left-top $p\times p$ submatrix of an $n\times n$ Haar-invariant unitary matrix. Suppose there exist two constants $0<h_1<h_2<1$ such that $h_1<\frac{p}{n}<h_2\mathrm{.}$ Then, $$\underset{x\in \mathbb{R}}{\sup }\mathrm{\mid }\mathbb{P}(X_n\le x)-e^{-e^{-x}}\mathrm{\mid }=\frac{(\log \log n{)}^2}{2e\log n}(1+o(1))$$ and further $$W_1(\mathcal{L}(X_n),\mathrm{\Lambda })=\frac{(\log \log n{)}^2}{2\log n}(1+o(1))$$ for $n$ large enough. Here, $\mathrm{\Lambda }$ is the Gumbel distribution and $\mathcal{L}(X_n)$ is the distribution of $X_n$ with $X_n$ being some rescaled version of ${\max }_{1\le i\le p}\mathrm{\mid }{z}_i\mathrm{\mid },$ the spectral radius of $A\mathrm{.}$