Fujii's conjecture on Gram interval statistics

Let $U(N)$ be the group of $N\times N$ unitary matrices, and let $J\subset[-\pi,\pi)$ be an interval of length $2\pi/N$. Define $E_{U(N)}(k,J)$ to be the probability that a random unitary matrix has exactly $k$ eigenvalues in $J$. Let $G_{0,M}(k)$ denote the proportion of Gram intervals containing exactly $k$ zeros in the relevant range. Fujii's conjecture. For every $k\in\mathbb N\cup\{0\}$,

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

The conjecture proposes a random-matrix model for Gram's Law by identifying the limiting eigenvalue-count distribution with the limiting distribution of zeros among Gram intervals. The source attributes the proposal to A. Fujii and does not state a resolution.