Gonek–Ki conjecture for zeros of real and imaginary parts of the zeta function

Fix $\theta\in[0,2\pi)$, let $a=1/2-c/\log T$ with $0<c=o(\log T)$, and let $A>0$ be arbitrarily large but fixed. Let $f_a(w,\theta)$ be the function in the source, and let $\Gamma(T)$ consist of the zeros of $f_a(w,\theta)$ on the line $\operatorname{Re}w=0$. Gonek–Ki conjecture. Uniformly for $0\leq|\alpha|\leq A$,

$$\mathfrak{F}_\Gamma(\alpha,T)=(2+o(1))T^{-4|\alpha|}\log T+\frac{\min(2|\alpha|,1)}{(a+1/2)(3-2a)}T^{(4a-2)|\alpha|}+o(1).$$

This conjecture predicts the relevant form factor for zeros of the family $f_a(w,\theta)$ along the imaginary axis. The paper explains that its bounds show the conjectured expression cannot hold in certain regimes, so the conjecture is refuted as stated.