Chandee's conjecture on shifted moments of the Riemann zeta function

Let $k\in\mathbb{N}$ and let ${\bf\alpha}=(\alpha_1,\alpha_2)$ satisfy the hypotheses stated for the shifted moments: the shifts are real-valued functions of $T$, are $o(T)$, have the required limits involving $\alpha_i\log T$ and $(\alpha_i-\alpha_j)\log T$, and satisfy $\alpha_1\ne\alpha_2$ with $\alpha_1-\alpha_2=O(1)$. Define

$$I_k(T,\alpha_1,\alpha_2)=\int_0^T\left|\zeta\left(\frac12+\mathrm{i}(t+\alpha_1)\right)\right|^k\left|\zeta\left(\frac12+\mathrm{i}(t+\alpha_2)\right)\right|^k\,dt.$$

Chandee's conjecture. One has

$$I_k(T,\alpha_1,\alpha_2)\begin{cases} \asymp_k T(\log T)^{k^2},&\text{if }\displaystyle\lim_{T\to\infty}|\alpha_1-\alpha_2|\log T=0,\\ \asymp_{k,c}T(\log T)^{k^2},&\text{if }\displaystyle\lim_{T\to\infty}|\alpha_1-\alpha_2|\log T=c\ne0,\\ \asymp_k T\left(\dfrac{\log T}{|\alpha_1-\alpha_2|}\right)^{k^2/2},&\text{if }\displaystyle\lim_{T\to\infty}|\alpha_1-\alpha_2|\log T=\infty. \end{cases}$$

This conjecture describes the transition in the size of two shifted zeta moments according to whether the shifts are closer than, comparable to, or farther apart than the scale $1/\log T$. It is presented as Chandee's conjecture based on the Keating–Snaith random matrix model; the paper proves related upper and lower bounds, but does not resolve the conjecture.