Conditional displaced fourth-moment conjecture for the Riemann zeta function

Let $N(T)$ denote the number of non-trivial zeros $\tfrac12+{\mathrm{i}}\gamma_n$ with $0<\gamma_n\leq T$, and set

$$L=\frac{1}{2\pi}\log\frac{T}{2\pi}.$$

Let $F_2$ be the function specified by the paper's formula for $F_2(2x)$. Conditional displaced fourth-moment conjecture. Assuming that the parameter choice $\eta=1$ is permissible in the cited theorem of Conrey, Ghosh and Gonek, as $T\to\infty$,

$$\frac{1}{N(T)}\sum_{0<\gamma_n\leq T}\left|\zeta\left(\tfrac12+{\mathrm{i}}\left(\gamma_n+\frac{\alpha}{L}\right)\right)\right|^4\sim\frac{1}{2\pi^2}F_2(2\pi\alpha)\left(\log\frac{T}{2\pi}\right)^4.$$

This is presented as the $k=2$ specialization of the displaced discrete-moment conjecture and is conditional because the theorem used does not allow $\eta=1$ as stated.