Conjecture on conic loci of the Gergonne and Nagel points

Let $\mathcal{L}_7$ and $\mathcal{L}_8$ denote the loci of the Gergonne point $X_7$ and Nagel point $X_8$, respectively, in a Poncelet triangle family. The pair of Poncelet conics may be confocal, or the caustic may be a circle; in the latter case, the incenter $X_1$ is stationary. Conic-locus conjecture for $X_7$ and $X_8$. $\mathcal{L}_7$ and/or $\mathcal{L}_8$ are conics if and only if either the Poncelet conic pair is confocal or the caustic is a circle. This is an experimental conjecture based on ten Poncelet families, and the source gives no proof or resolution.