Alternative SU(N) conjecture for Gram interval statistics

Let $E_{SU(N)}(k,\mathcal J)$ denote the probability that a random special unitary matrix has exactly $k$ eigenvalues in the corresponding $SU(N)$ Gram interval $\mathcal J$, and let $G_{0,M}(k)$ denote the proportion of Gram intervals containing exactly $k$ zeros in the relevant range. Alternative SU(N) conjecture. For every $k\in\mathbb N\cup\{0\}$,

$$\lim_{N\to\infty}E_{SU(N)}(k,\mathcal J)=\lim_{M\to\infty}G_{0,M}(k).$$

This conjecture is proposed as an alternative to the $U(N)$ model because the $SU(N)$ probabilities appear to approximate low-height Gram-interval statistics with the same rate of convergence, although the source does not establish the common limiting value.