Positive links with arrangements of pseudocircles as shadows

An arrangement of pseudocircles $\mathcal{A}$ is a collection of Jordan curves in the plane that pairwise intersect (transversally) at exactly two points. How many non-equivalent links have $\mathcal{A}$ as their shadow? Motivated by this question, we study the number of non-equivalent positive oriented links that have an arrangement of pseudocircles as their shadow. We give sharp estimates on this number when $\mathcal{A}$ is one of the three unavoidable arrangements of pseudocircles.