Smoothed weighted one-level density conjecture for the Riemann zeta-function

Let $N_f(t):=\sum_\gamma f\left((\gamma-t)/(2\pi/\log T)\right)$, where the sum is over the non-trivial zeros of the Riemann zeta-function. Let $k\in\mathbb N$, let $f$ be even with $\hat f\in\mathcal{C}^{\infty}_c(\mathbb R)$, and let $\phi$ be smooth with compact support in $\mathbb R_{>0}$. Define its Mellin transform by

$$\widetilde\phi(s):=\int_{\mathbb R}\phi(x)x^{s-1}\,dx.$$

Smoothed weighted one-level density conjecture. As $T\to\infty$,

$$\frac{1}{c_kT(\log T)^{k^2}}\int_{\mathbb R}N_f(t)|\zeta(\tfrac12+it)|^{2k}\phi\left(\frac{t}{T}\right)\,dt=\widetilde\phi(1)\int_{\mathbb R}f(x)W_U^k(x)\,dx+O((\log T)^{-1}).$$

This is the naturally smoothed version of the weighted one-level density conjecture and is expected to hold for all positive integers $k$; no asymptotic formula is known for the associated moments over zeros when $k>1$.