The shifted moments conjecture for the Riemann zeta-function

Let $k\in\mathbb N$, let $Y=Y(T)$, and let $R$ be a piecewise smooth positively oriented path encircling $0$ and $-s$. Shifted moments conjecture. For $T\geq1$,

$$\frac1T\int_0^T\zeta\left(\frac12+it+s-z\right)^k\zeta\left(\frac12-it+z\right)^kdt=\frac{a_k}{k!(2\pi i)^k}\int_{R^k}T^{\sum_{j=1}^k w_j}\prod_{i=1}^k w_i^{-k}(w_i+s)^{-k}\prod_{i\ne j}(w_i-w_j)\,d\boldsymbol w+O_k\left((\log T)^{k^2-1+\varepsilon}(1+\min(|s|,1)\log T)^{-k^2/2+1}\right).$$

This holds for $|\Re z|,|\Re s|\leq3/\log T$ and $|\Im z|,|\Im s|\ll Y$. It is a uniform shifted-moment hypothesis used to derive the smoothness formula; the paper notes that the admissible range is conjectural and that the needed uniformity is known for $k\leq2$.