Möbius-twisted Elliott–Halberstam conjecture

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Euler totient function. Möbius-twisted Elliott–Halberstam conjecture. For every $A>0$,

$$\sum_{q\leq N^{\theta}}\max_{y<N}\max_{(a,q)=1}\left|\sum_{\substack{n\leq y\\ n\equiv a\,\operatorname{mod} q}}\Lambda(n)\mu(N-n)-\frac{1}{\varphi(q)}\sum_{n\leq y}\Lambda(n)\mu(N-n)\right|\ll_A\frac{N}{\log(N)^A}.$$

This conjecture is designed for the Goldbach problem, where the Möbius factor is evaluated at the complementary variable $N-n$. Its range beyond the Bombieri–Vinogradov threshold is open and would have strong consequences for binary Goldbach-type questions.