Moses' pencil axis-ratio invariance conjecture for 3-periodic circumellipses

Let a 3-periodic be a member of the one-parameter family of triangular orbits in an elliptic billiard, and let Moses' pencil be the pencil of circumconics whose centers lie on the Feuerbach circumhyperbola of the medial triangle. A circumellipse in this pencil has two axis lengths.

Moses' pencil conjecture. Over the family of 3-periodics, all circumellipses in Moses' pencil conserve the ratio of their axes.

This is an experimentally observed invariant for the circumellipses in the parallel-axis pencil; the source does not provide a proof or resolution.