Hejhal's triple correlation conjecture for Riemann zeros

Let $\gamma,\gamma',\gamma”$ range over ordinates of nontrivial zeros of the Riemann zeta function, put $\widetilde\gamma=(\log T)/(2\pi)\,\gamma$, and let $\hat r$ be the relevant two-dimensional Fourier transform. Define

$$H(a,b)=H_{\delta}(a,b)+H_*(a,b),$$

where

$$H_{\delta}(a,b)=\delta(a)\delta(b)+\delta(a)\min\{|b|,1\}+\delta(b)\min\{|a|,1\}+\delta(a+b)\min\{|a|,1\text{ }\},$$ $$H_*(a,b)=2G(a,b)+\min\{|a|,1\}+\min\{|b|,1\}+\min\{|a+b|,1\}-2,$$

and $G(a,b)=\max\{(2-|a|-|b|-|a+b|)/2,0\}$. Hejhal's triple correlation conjecture. For every continuous integrable function $r$ such that $\hat r$ is Lipschitz continuous and integrable,

$$\left(\frac{T\log T}{2\pi}\right)^{-1}\sum_{T\leq\gamma,\gamma',\gamma”\leq2T}r(\widetilde\gamma-\widetilde\gamma',\widetilde\gamma-\widetilde\gamma”)=\int_{\mathbb R}\int_{\mathbb R}\hat r(a,b)H(a,b)\,da\,db+O\left(\frac1{\log T}\right).$$

This is the three-point analogue of Montgomery's pair correlation conjecture and is motivated by the sine-kernel correlations of random unitary matrices. The source notes that restricted-support versions of general $n$-point correlation results are known, while this full-strength formulation remains conjectural.