Barban–Davenport–Halberstam conjecture for primes in short intervals and arithmetic progressions

Let $\theta(x;q,a)$ denote the weighted count of primes up to $x$ in the reduced residue class $a$ modulo $q$:

$$\theta(x;q,a)=\sum_{\substack{p\le x\p\equiv a\pmod q}}\log p.$$

Let $0<\eta\le 1$ and $\beta>0$ be real numbers. Suppose that $X$, $Y$, and $Q$ are positive real numbers satisfying

$$X^\eta\le Y\le X,\qquad \frac{Y}{(\log X)^\beta}\le Q\le Y.$$

Barban–Davenport–Halberstam conjecture. Under these conditions,

$$\sum_{q\le Q}\sum_{\substack{1\le a\le q\\(a,q)=1}}\left|\theta(X+Y;q,a)-\theta(X;q,a)-\frac{Y}{\phi(q)}\right|^2\ll_{\eta,\beta}YQ\log X.$$

This is a short-interval analogue of the Barban–Davenport–Halberstam theorem for primes in arithmetic progressions. The statement is used as an assumed distributional hypothesis for primes and is not established in the source excerpt.