Conjecture on stationary orthocenters in Poncelet triangle families

Let $X_4$ be the orthocenter of a Poncelet triangle family interscribed between two conics. The configurations called focal-$X_4$, MacBeath, and Dual are the three configurations identified in the source. Stationary-orthocenter conjecture. A Poncelet triangle family maintains $X_4$ stationary only if the conics are configured as focal-$X_4$, MacBeath, or Dual. The source exhibits these configurations and states the classification as a conjecture; no resolution is provided.