Diagonal Möbius–logarithmic Elliott–Halberstam conjecture

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

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

This is a diagonal, in some sense weaker, variant of the Möbius-twisted Elliott–Halberstam conjecture, tailored to the symmetric weighted averages studied in the paper. Its validity is open.