“…Following [20, Lemma 2] or [19,Lemma 3.2], by "shrinking and straightening" α or β, we may assume that D has only one maximal point with respect to h. Since P is a thin level 2-sphere, we may assume without loss of generality that β has a minimal point. Let P be a "thinnest" level 2-sphere for β, that is, either a thin level 2-sphere between a maximal point p and a minimal point q of β or possibly parallel to P such that it has the minimal number of intersection with β in the region…”