Classification of Poncelet triangle families with stationary orthocenter

Let a Poncelet triangle family consist of triangles simultaneously inscribed in and circumscribed about a pair of conics, and let $X_4$ denote the orthocenter. The conics may be configured as focal-$X_4$, MacBeath, or Dual.

Stationary-orthocenter classification conjecture. A Poncelet triangle family maintains $X_4$ stationary only if the conics are configured as focal-$X_4$, MacBeath, or Dual.

This conjecture proposes a classification of Poncelet triangle families with stationary orthocenter, beyond the explicitly identified focal-$X_4$, MacBeath, and Dual families. The source provides no resolution or further evidence of its status.