  Let $T$ be an $n\times n$ truncation of an $(n+\alpha)\times (n+\alpha)$ Haar
distributed unitary matrix. We consider the disk counting statistics of the
eigenvalues of $T$. We prove that as $n\to + \infty$ with $\alpha$ fixed, the
associated moment generating function enjoys asymptotics of the form
\begin{align*} \exp \big( C_{1} n + C_{2} + o(1) \big), \end{align*} where the
constants $C_{1}$ and $C_{2}$ are given in terms of the incomplete Gamma
function. Our proof uses the uniform asymptotics of the incomplete Beta
function.
