How tall must a parabola be in order to roll and completely flip over? (Made in Blender 2.91)

This is a simulation built in Blender 2.91. Several instances of x^2 are shown but chopped off at varying heights. Initially they are all leaning ~1 degree to the right, and then gravity takes over. The two points are at their respective parabola's center of mass.

The parabolas roll friction free. Can you work out the precise minimum height needed to flip over on a non-slip surface? It involves center of mass, arc length, and a lot of patience.

Forgive my n00b blender skills. This is my first physics simulation. I don't know why some of the parabolas lag behind the group. I played around with the simulation settings a bunch and couldn't figure it out. Also this simulation is affected by the "curvature" of the parabola, which is actually just very short line segments. I'm sure the accuracy of the simulation could be improved by a different meshing technique, but I'm just learning don't shoot me.