The divisor correlation conjecture for shifted divisor functions

Let $k,l\geqslant 2$ be natural numbers, let $h\in\mathbb{Z}\setminus\{0\}$, and let $d_k(n)$ and $d_l(n)$ denote the $k$-fold and $l$-fold divisor functions. Divisor correlation conjecture. As $X\to\infty$, one has

$$\sum_{X<n\leqslant 2X} d_k(n)d_l(n+h)=P_{k,l,h}(\log X)X+O(X^{1/2+o(1)}),$$

for some polynomial $P_{k,l,h}$ of degree $k+l-2$. This is the divisor-function analogue of the Hardy–Littlewood prime tuple conjecture; the cited source records it as a conjectural asymptotic, while the present paper proves related asymptotics only in shorter intervals and for almost all parameters.