The least prime number represented by a binary quadratic form

Let $D 0$ is an absolute positive constant independent of $D$. More generally, let $K$ be a bounded degree number field over $\mathbb{Q}$ with the discriminant $D_K$ and the class number $h_K.$ We conjecture that a positive proportion of the ideal classes of $K$ contain a prime ideal with a norm less than $h_K\log(|D_K|)$.