Exponential limit conjecture for the rescaled Schur off-diagonal entry

Let $X$ be a random matrix satisfying the assumptions of Theorem, and let $T$ be its Schur form. Define $Y=n|t_{12}|^2$, where $t_{12}$ is the relevant off-diagonal entry of $T$, and let $\mathbb{E}Y$ denote its expectation. Exponential limit conjecture. The sequence

$$\frac{Y}{\mathbb{E}Y}$$

converges in distribution, as $n\to\infty$, to the exponential law of parameter one. This conjecture extends the corresponding Gaussian behavior from the Ginibre ensemble to the general setting considered in Theorem; its resolution is not specified in the supplied text.