The CFKRS moment conjecture for the Riemann zeta function

The Riemann zeta function is defined by $[... ELLIPSIZATION...$ and continued meromorphically to the complex plane. For complex shifts $[... ELLIPSIZATION...$, define the $2k$-point autocorrelation integral by the left-hand side below, and let $W_k$ be the explicit expression involving the permutation set $[... ELLIPSIZATION...$ and Euler product $A_k$ given in the statement. CFKRS moment conjecture. For suitable shifts and $[... ELLIPSIZATION...$, the autocorrelation is given by

$$\int_0^T \zeta(\tfrac12+\alpha_1+it)\cdots \zeta(\tfrac12+\alpha_k+it)\zeta(\tfrac12-\alpha_{k+1}-it)\cdots\zeta(\tfrac12-\alpha_{2k}-it)\,dt =\int_0^T W_k(t;\alpha_1,\ldots,\alpha_k;\alpha_{k+1},\ldots,\alpha_{2k})(1+O(t^{-1/2+\epsilon}))\,dt.$$

Here $W_k$ is the expression displayed in the source, with $\Xi$ the set of $\binom{2k}{k}$ block-preserving permutations and $A_k$ the displayed Euler product. This is a conjectural asymptotic formula for zeta moments; the source presents it as the main comparison with random matrix theory and does not provide a resolution.