Generalized Minkowski bound for prime ideals in ideal class groups

Let $K$) be a number field of bounded degree $n$ over $\mathbb{Q}$, with discriminant $D_K$ and class number $h_K$. An ideal class is said to contain a prime ideal if some prime ideal in that class has norm below the specified bound. Generalized Minkowski bound. Almost all ideal classes in the ideal class group of $K$ contain a prime ideal with norm less than

$$h_K\log(|D_K|)^A$$

for some $A>0$. The conjecture generalizes the heuristic bound for imaginary quadratic fields; under the Brauer–Siegel theorem and GRH one has $h_K\ll\sqrt{|D_K|}\log(|D_K|)^\epsilon$, while the asserted bound for all number fields remains open.