Before revision:  In this paper we answer this question , under an even more general definition of an arrangement, in which the pseudocircles in the collection are not required to intersect each other, or that the intersections are crossings: it suffices to have one pseudocircle that intersects all other pseudocircles in the collection. We show that under this more general notion, an arrangement of pseudocircles is embeddable into \Sigma_g if and only if all of its subarrangements of size at most 4g+ 5 areembeddable into \Sigma_g, and that this can be improved to 4g+4 under the concept of an arrangement used by Ortner.
After revision:  In this paper we answer this question positively: An arrangement of pseudocircles is embeddable into \Sigma_g if and only if all of its subarrangements of size at most 4g+ 5 areembeddable into \Sigma_g, and that this can be improved to 4g+4 under the concept of an arrangement used by Ortner.