Strict ordering conjecture for billiard winding numbers in ellipsoids

Let $m_0,\ldots,m_{n-1}$ be the winding numbers of a periodic billiard trajectory inside an ellipsoid, as defined by the oscillations of its elliptic coordinates. Winding-number ordering conjecture. Winding numbers are always ordered in a strict decreasing way:

$$2 \leq m_{n-1} < \cdots < m_1 < m_0.$$

This conjecture was stated in earlier work and tested numerically. The supplied text gives no resolution, so its general status remains open.