Before revision: This paper proves a conjecture from LO17 : A nearly flat, acutely triangulated convex cap C  has an edge-unfolding to a non-overlapping polygon in the plane.
After revision: This paper proves a conjecture from LO17 : A nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane.