Pair-correlation conjecture for zeros of Dirichlet L-functions

For each Dirichlet character $\chi$ modulo $q$, let $\gamma_\chi$ and $\gamma'_\chi$ denote ordinates of nontrivial zeros of $L(s,\chi)$, counted with multiplicity. Fix $C>0$. Pair-correlation conjecture. There exists a bounded function $W(t)\geq 0$ such that, whenever $C^{-1}\leq S\leq\log q$ and $1\leq U\leq C\log Q$,

$$\sum_{\chi\bmod q}\#\left\{0\leq\gamma_\chi,\gamma'_\chi\leq U:0<|\gamma_\chi-\gamma'_\chi|\leq\frac{S}{\log q}\right\} \ll_C \phi(q)U\log(qU)\int_0^S W(t)\,dt.$$

This conjecture is introduced to control the contribution of pairs of distinct zeros in the moment calculation. Its resolution would support the paper's probabilistic predictions for the variance $V(x;q)$, but the supplied text gives no resolution.