Generalized Elliott–Halberstam conjecture for shifted convolutions

Let $x\geq2$, $q\in\mathbb{N}$, and $a,h\in\mathbb{Z}$ with $(a,q)=1$. Define

$$E_2(x;q,a,h)=\sum_{\substack{n\leq x\ \equiv a\pmod q}}\Lambda(n)\Lambda(n+h)-\frac{\mathbf{1}_{(a(a+h),q)=1}}{\varphi(q)}\mathfrak{S}(h)x,$$

where $\mathbf{1}_{(a(a+h),q)=1}$ equals $1$ when both $a$ and $a+h$ are coprime to $q$, and equals $0$ otherwise. Generalized Elliott–Halberstam conjecture for shifted convolutions (GEH-2). For every $0<\vartheta<2$, every fixed $h\neq0$, and every $A>0$,

$$\sum_{q\leq x^\vartheta}\max_{(a,q)=1}|E_2(x;q,a,h)|\ll\frac{x}{(\log x)^A},$$

with implied constant depending only on $\vartheta$, $h$, and $A$. This extends Elliott–Halberstam from individual primes to shifted correlations of the von Mangoldt function and is presented as implying the twin prime conjecture; it remains unproved.