Conjecture on the circumcircle envelope of Poncelet triangles

Let $\mathcal{T}$ be the family of Poncelet triangles, and let $\mathcal{E}_c$ be the inner Poncelet caustic. Consider the envelope of the circumcircle as the triangles range over $\mathcal{T}$. Circumcircle-envelope conjecture. Either component of this envelope is a conic if and only if $\mathcal{E}_c$ is a circle; in that case both components are circles. The circular-caustic case is established in the source, while the assertion for a general caustic is supported there by experimental evidence.