The bic-III non-conic conjecture for triangle-center loci

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with each side tangent to a distinct in-pencil circle, and assume all four in-pencil circles are distinct. Let $X$ be a triangle center. The bic-III non-conic conjecture. Over the bic-III family, the locus of a triangle center is never a conic. The claim is based on experimental evidence for the bic-III construction; the source gives no proof or resolution.