Additive divisor conjecture with a power-saving error term

Let $k,\ell>1$ be natural numbers, let $x$ be large, and let $h$ satisfy $1\le h\le x^{1-\varepsilon}$. Let $D_{k,\ell}(x,h)=\sum_{n\le x}\tau_k(n)\tau_\ell(n+h)$, and let $q_{k,\ell}(t,h)$, $E_{k,\ell}(x,h)$, $C_{k,\ell}$, and $f_{k,\ell}(h)$ be the quantities defined in the source. Additive divisor conjecture. There exists a positive constant $\vartheta_{k,\ell}\in[\tfrac12,1)$ such that

$$D_{k,\ell}(x,h)=\int_0^x q_{k,\ell}(t,h)\,dt+E_{k,\ell}(x,h),$$

and, for every $\varepsilon>0$,

$$E_{k,\ell}(x,h)\ll x^{\vartheta_{k,\ell}+\varepsilon}$$

uniformly for $1\le h\le x^{1-\varepsilon}$. Moreover, the coefficient of $\log(t)\log(t+h)$ in $q_{k,\ell}(t,h)$ is $\mathcal{D}_{k,\ell}(0,0)=C_{k,\ell}f_{k,\ell}(h)$. This conjecture gives a refined asymptotic framework for shifted divisor correlations, including an explicit local leading coefficient and a conjectural error exponent; the source does not resolve the conjectured bound.