Edited after mistake in the first version.

It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a set of asymptotic density one (Selberg's result is actually more precise: one can take $x+O(f(c) \log^2 x)$ where $f(x)$ is any function that tends to $x+\infty$ with $x$). Here a set of density one is a subset $S \in \mathbb R_+$ such that $\mu(S \cap [0,y])/y \longrightarrow 1$ when $y \longrightarrow \infty$.

I have heard that this result has been generalized for primes in arithmetic progressions under GRH. I would like to know if more generally that result has been generalized for primes in a Frobenian set? More precisely,

Given a fixed Galois number field $L/\mathbb Q$, $G=Gal(L/\mathbb Q)$, $C$ a conjugacy class in $G$, is it true that for every $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ such that $Frob_{p,Gal(L/\mathbb Q)} \in C$ for every $x$ in a set of asymptotic density one (under GRH, and Artin's conjecture if you wish)?

I am looking for any reference discussing that question... If there is none available (as it seems at first glance according to my looking in mathscinet), I would be also interested to the clearest references you know treating the case of arithmetic sequence. In any case, thanks...