Axis-ratio invariance for Circumellipses in Moses' pencil

Let a triangle vary through a family of 3-periodic orbits, and consider the Circumellipses in Moses' pencil, namely the Circumellipses whose centers lie on the Feuerbach Circumhyperbola $F_{med}$ of the Medial Triangle. Axis-ratio invariance conjecture. Over the family of 3-periodics, all Circumellipses in Moses' pencil conserve the ratio of their axes. This has been observed experimentally; the source provides no proof or resolution, so the conjecture remains open.