Microscopic zeta-function process convergence to the limiting characteristic function

Let $\omega$ be uniform on $[0,1]$, let $T>0$ tend to infinity, and let $\xi_{\infty}$ be the limiting random analytic function arising from the microscopic scaling of characteristic polynomials of Haar-distributed unitary matrices. For $z\in\mathbb C$, consider the randomized, renormalized zeta function

$$F_T(z):=\frac{\zeta\left(\frac12+iT\omega-\frac{2\pi i z}{\log T}\right)}{\zeta\left(\frac12+iT\omega\right)}.$$

Microscopic zeta-function convergence conjecture. In law, uniformly for $z$ on every compact subset of $\mathbb C$,

$$\bigl(F_T(z);z\in\mathbb C\bigr)\underset{T\to\infty}{\longrightarrow}\bigl(\xi_{\infty}(z);z\in\mathbb C\bigr).$$

Moreover, on compact sets bounded away from the real line, the logarithmic derivatives should satisfy

$$\left(\frac{-2\pi i}{\log T}\frac{\zeta'}{\zeta}\left(\frac12+iT\omega-\frac{2\pi i z}{\log T}\right);z\in\mathbb C\right)\underset{T\to\infty}{\longrightarrow}\left(\frac{\xi_{\infty}'}{\xi_{\infty}}(z);z\in\mathbb C\right).$$

This is the source's central process-level conjecture linking the microscopic zeta function to the limiting random analytic function. It would imply corresponding predictions for microscopic zeros and logarithmic derivatives, but no resolution is stated.