The GUE hypothesis for local correlations of zeta zeros

Let $\{\sigma_j+i\gamma_j\}_{j\in\mathbb{Z}}$ be the non-trivial zeros of the Riemann zeta-function, ordered by their ordinates, and let

$$\widetilde{\gamma}_j=\gamma_j\frac{\log|\gamma_j|}{2\pi}$$

be the normalized zeros. Define

$$S(x)=\begin{cases}\frac{\sin\pi x}{\pi x}&\text{if }x\ne 0,\\1&\text{if }x=0.\end{cases}$$

GUE hypothesis. For $n\geq 1$ and any $\varphi\in\mathcal{S}(\mathbb{R}^n)$,

$$\lim_{T\to\infty}\frac{1}{T}\int_T^{2T}\sum_{\substack{j_1,\ldots,j_n\in\mathbb{Z}^n\mathrm{distinct}}}\varphi(\widetilde{\gamma}_{j_1}-t,\ldots,\widetilde{\gamma}_{j_n}-t)\,dt=\int_{\mathbb{R}^n}\varphi(x)\det_{1\leq i\leq j\leq n}\big[S(x_i-x_j)\big]~d^nx.$$

Equivalently, if $\mathcal{X}_T$ is the point process whose configurations are $(\widetilde{\gamma}_j-t)_{j\in\mathbb{Z}}$, with $t$ chosen uniformly from $[T,2T]$, then $\mathcal{X}_T$ should tend in distribution to the sine-kernel process as $T\to\infty$. The hypothesis is widely believed and is known for some classes of test functions, but the full assertion remains open; it emerged from work of Montgomery and has numerical support from Odlyzko.