Invariant powers of fixed points for circumcircles and Euler circles

Consider a non-concentric, non-axis-aligned pair of ellipses admitting Poncelet 3-periodics. For each 3-periodic, let its circumcircle and Euler circle be the circles associated with the triangle. Fixed-point power conjecture. There is a fixed point $P_3$ whose power with respect to the circumcircle is invariant over the family, and there is a fixed point $P_5$ whose power with respect to the Euler circle is invariant over the family. This is presented as an experimental phenomenon for pairs in general position, and the source gives no proof or resolution.