Suppose that $O$ is a uniformly random orthogonal matrix (w.r.t. Haar measure) and $X$ be its top-left $k\times k$ block.

There have been some literature studying the distribution of eigenvalues of $X$. Actually, there are more complete results for submatrix of random unitary matrix, which says that the eigenvalues are distributed like the eigenvalues of Gaussian random matrix (properly normalized). Presumably similar results hold for submatrix of random orthogonal matrix.

But I have not seen much result regarding the singular values of submatrix of orthogonal matrix. Jiang has a paper that studies the singular valued distribution of $X$ when $k=o(\sqrt{n})$, basically based on the fact (his earlier paper) that when $k=o(\sqrt{n})$, $X$ is statistically close to a Gaussian random matrix.

My question is, what happens when $k>\sqrt{n}$? Put the distribution of singular values of $X$ aside, do we know anything about the operator norm of $\|X\|$? Is it concentrated? Are there such results in the literature?