The arithmetic correlations conjecture for convolutions of divisor sums

Let $\alpha,\beta$ be complex shifts and let $\gamma,\delta$ be shifts for which the quantities below are defined. Define the arithmetic coefficients $I_{\alpha,\gamma}(n)$ and $I_{\beta,\delta}(n)$ by

$$\sum_{n=1}^\infty\frac{I_{\alpha,\gamma}(n)}{n^s}=\frac{\zeta(s+\alpha)}{\zeta(s+\gamma)},\qquad \sum_{n=1}^\infty\frac{I_{\beta,\delta}(n)}{n^s}=\frac{\zeta(s+\beta)}{\zeta(s+\delta)}.$$

For the Ramanujan sums $r_q(h)$ and the arithmetic factors $F_{\alpha,\gamma}(q)$ and $F_{\beta,\delta}(q)$ occurring in the conjectural main term, set

$$m(x,h)=\frac{1}{\zeta(1+\gamma-\alpha)\zeta(1+\delta-\beta)}\sum_{q=1}^\infty\frac{r_q(h)F_{\alpha,\gamma}(q)F_{\beta,\delta}(q)}{q^2}\frac{x^{1-\alpha-\beta}}{1-\alpha-\beta}.$$

Arithmetic correlations conjecture. There are numbers $\phi<1$ and $\psi>0$ such that

$$\sum_{n\le x}I_{\alpha,\gamma}(n)I_{\beta,\delta}(n+h)=m(x,h)+O(x^\phi)$$

uniformly for $h\ll x^\psi$. This conjecture supplies the arithmetic input used in the paper's main theorem and is intended to explain the correlations underlying the ratios conjecture. The source does not state a resolution, so it remains open.