On convex spiral equicoverings of masses

Convex spiral equicoverings were recently introduced by Espinosa-García, Martínez-Sandoval and Roldán-Pensado. They left open the question of whether every planar mass admits a convex $(3k,k+1)$-spiral equicovering. In this paper we give an affirmative answer to this question. To be precise, we prove the following: Given an integer $k\ge 2$, for every planar mass there is a fan consisting of $3k$ equal-mass sectors such that the union of every $k+1$ consecutive sectors is convex.