On the Montgomery­-Hooley theorem in short intervals

We study a short-interval version of a result due to Montgomery and Hooley. Write \[ S(x,h,Q) = \sum_{q \le Q} \sum_{\substack{a = 1 \\ (a, q) = 1}}^q \left\vert \psi(x + h; q, a) - \psi(x; q, a) - \frac h {\varphi(q)} \right\vert^2 \] and $\kappa = 1 + \gamma + \log 2\pi + \sum_p (\log p) / p (p - 1)$. Denote the expected main term by $M(x, h, Q) = h Q \log (x Q / h) + (x + h) Q \log(1 + h / x) - \kappa h Q$. Let $\epsilon$, $A > 0$ be arbitrary, $x^{7/12+\epsilon} \le h \le x$ and $Q \le h$. There exists a positive constant $c_1$ such that \[ S(x, h, Q) - M(X, h, Q) \ll h^{1/2} Q^{3/2} \exp \left(-c_1\frac{(\log 2h/Q)^{3/5}}{(\log\log 3h/Q)^{1/5}} \right) + h^2 \log^{-A} x. \] Now assume \emph{GRH} and let $\epsilon > 0$, $x^{1/2+\epsilon} \le h \le x$ and $Q \le h$. There exists a positive constant $c_2$ such that \[ S(x, h, Q) - M(x, h, Q) \ll \Bigl( \frac hQ \Bigr)^{1/4+\epsilon} Q^2 + h x^{1/2} \log^{c_2} x. \]