The bic-III convexity conjecture for the incenter locus

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with $P_1P_2$ and $P_1P_3$ tangent to distinct circles $\mathcal{C}'$ and $\mathcal{C}”$ in the pencil of $\mathcal{C}$ and $\mathcal{C}'$, and with $P_2P_3$ tangent to a third in-pencil circle $\mathcal{C}”'$. Assume all four in-pencil circles are distinct. The bic-III convexity conjecture. Over the bic-III family, the locus of $X_1$ is a convex curve. Here $X_1$ denotes the incenter. The claim is presented as experimental evidence, and no proof or resolution is supplied.