Ellipse-locus conjecture for triangle centers on focus-mounted triangles

Let $\mathcal{E}$ be an ellipse with foci $f_1$ and $f_2$, and let $P(t)$ sweep $\mathcal{E}$. For the triangle $\mathcal{T}_f(t)=f_1f_2P(t)$, denote by $X_k$ the $k$th triangle center, and let $X_1$ and $X_2$ be the first and second triangle centers. A point $X_k$ is a fixed affine combination of $X_1$ and $X_2$ if there are constants independent of $t$ such that $X_k=\lambda X_1+(1-\lambda)X_2$. Ellipse-locus conjecture. The locus of $X_k$ over $\mathcal{T}_f(t)=f_1f_2P(t)$ is an ellipse if and only if $X_k$ is a fixed affine combination of $X_1$ and $X_2$. Numerical analysis found no counterexample among the more than 38,000 listed centers that are not on the $X_1X_2$ line, while the proposition preceding the conjecture identifies several centers whose loci are ellipses. The claim remains unproved in the supplied text.