Smooth-variation conjecture for pair correlation of Selberg-class L-functions

Let $F\in\mathcal S$ and let $\mathcal D_F(x,T)$ be the normalized pair-correlation function defined above. Smooth-variation conjecture. For every $A>0$, there exists $c>0$ such that $\mathcal D_F(x,T)\ll1$ uniformly in $x$ and $T$, and

$$\left|\mathcal D_F(x+\delta x,T)+\mathcal D_F(x-\delta x,T)-2\mathcal D_F(x,T)\right|=o\left(\frac{T}{\log T}\right)$$

uniformly for $T\le x\le T^A$ and $0\le\delta\le(\log T)^{c-1}$. This conjecture is proposed as the general-$L$-function analogue of Chan's smoothly varying pair-correlation behavior through the transition zone. It is not proved in the paper and remains open.