Conjecture on the envelope of the circumcircle-incenter line for Poncelet triangles

Let $\mathcal{T}_{\circ}$ be the Poncelet family whose circumcircle is constrained by the circular caustic, and let $\Gamma$ denote the line associated with the circumcircle-incenter construction described in the source. Envelope-conic conjecture. Over $\mathcal{T}_{\circ}$, the envelope of $\Gamma$ is a conic if and only if the Poncelet caustic is a circle. The source presents this assertion as based on experimental evidence; in the equilateral specialization, the corresponding envelope is described separately as a parabola.