Keating–Rodgers–Roditty-Gershon–Rudnick conjecture for divisor-function short intervals

Let $U_n$ be uniformly distributed on $\{n+1,\ldots,2n\}$, let $d_k(N)$ be the $k$-th divisor function, and set $\Delta_k(U_n)=d_k(U_n)-\mathbb{E}(d_k(U_n))$ and $\Delta_k(U_n,H)=\Delta_k(U_n+H)-\Delta_k(U_n)$. Define $\mathcal{I}_c(k)$ as the limiting function appearing in the cited function-field analogue, and set $\mathcal{P}_k(\delta)=(1-\delta)^{k^2-1}\mathcal{I}_{(1-\delta)^{-1}}(k)$. Keating–Rodgers–Roditty-Gershon–Rudnick conjecture. For $n\to+\infty$ and $\delta\in(0,1-k^{-1})$,

$$\mathbb{E}\left(\Delta_k(U_n,n^\delta)^2\right)\sim a_k\mathcal{P}_k(\delta)n^\delta(\log n)^{k^2-1},$$

where $a_k$ is the arithmetic factor defined above. This is a number-field conjecture motivated by a function-field calculation and concerns the variance of divisor-function differences in short intervals. The supplied passage gives no resolution status.