  We prove that every plane passing through the origin divides an embedded
compact free boundary minimal surface of the euclidean $3$-ball in exactly two
connected surfaces. We also show that if a region in the ball has mean convex
boundary and contains a nullhomologous diameter, then this region is a closed
halfball. Moreover, we prove the regularity at the corners of currents
minimizing a partially free boundary problem by following ideas by Gr\"uter and
Simon. Our first result gives evidence to a conjecture by Fraser and Li.
