Convexity and symmetry conjecture for the incenter locus

Let a pair of ellipses admit Poncelet 3-periodics, and let $X_1$ denote the incenter of a member of this family. If the pair is concentric at $O$, call $O$ the common center; otherwise, distinguish the two ellipse centers and the center of the locus diameter. Incenter-locus convexity and symmetry conjecture. The locus of $X_1$ is always convex. If the pair is concentric at $O$, the locus diameter passes through $O$ and the locus is four-fold symmetric about $O$. If the pair is non-concentric, the diameter does not pass through either ellipse center and the locus is asymmetric about the center of its diameter. The claim is based on the depicted locus calculations and experimental observations; a proof of the asserted convexity and symmetry properties is not provided.