  Consider a truncated circular unitary matrix which is a $p_n$ by $p_n$
submatrix of an $n$ by $n$ circular unitary matrix after deleting the last
$n-p_n$ columns and rows. Jiang and Qi \cite{JiangQi2017} and Gui and Qi
\cite{GQ2018} study the limiting distributions of the maximum absolute value of
the eigenvalues (known as spectral radius) of the truncated matrix. Some
limiting distributions for the spectral radius for the truncated circular
unitary matrix have been obtained under the following conditions: (1). $p_n/n$
is bounded away from $0$ and $1$; (2). $p_n\to\infty$ and $p_n/n\to 0$ as
$n\to\infty$; (3). $(n-p_n)/n\to 0$ and $(n-p_n)/(\log n)^3\to\infty$ as
$n\to\infty$; (4). $n-p_n\to\infty$ and $(n-p_n)/\log n\to 0$ as $n\to\infty$;
and (5). $n-p_n=k\ge 1$ is a fixed integer. The spectral radius converges in
distribution to the Gumbel distribution under the first four conditions and to
a reversed Weibull distribution under the fifth condition. Apparently, the
conditions above do not cover the case when $n-p_n$ is of order between $\log
n$ and $(\log n)^3$. In this paper, we prove that the spectral radius converges
in distribution to the Gumbel distribution as well in this case, as conjectured
by Gui and Qi \cite{GQ2018}.
