Exploiting experts' knowledge can significantly increase the quality of the Bayesian network (BN) structures produced by learning algorithms. However, in practice, experts may not be 100% confident about the opinions they provide. Worst, the latter can also be conflicting. Including such specific knowledge in learning algorithms is therefore complex. In the literature, there exist a few score-based algorithms that can exploit both data and the knowledge about the existence/absence of arcs in the BN. But, as far as we know, no constraint-based learning algorithm is capable of exploiting such knowledge. In this paper, we fill this gap by introducing the mathematical foundations for new independence tests including this kind of information. We provide a new constraint-based algorithm relying on these tests as well as experiments that highlight the robustness of our method and its benefits compared to other constraint-based learning algorithms.