Uniqueness of the conic-sweeping triangle center for inversive Poncelet triangles

Let $\mathcal{E}$ and $\mathcal{E}_c$ be nested ellipses admitting a family $\mathcal{T}$ of Poncelet triangles, and let $T=ABC$ be a triangle in $\mathcal{T}$. Let $T'=A'B'C'$ be obtained by inverting the vertices of $T$ with respect to a fixed circle $\mathcal{K}=(O,r)$, and let $X_3'$ denote the circumcenter of $T'$. In a generic Poncelet family, the uniqueness conjecture. the only triangle center of $T'$ whose locus is a conic is the circumcenter $X_3'$. This is suggested by experiments and complements the theorem that the locus of $X_3'$ is a conic; the claim concerns whether any other triangle center of the inversive Poncelet triangles can have a conic locus.