Can every inscribable 3-polytope be circumscribed while still preserving the maximum symmetry of its canonical form?

Question

According to Steinitz's theorem, it is possible to prove that every 3-connected graph G(P) represents the skeleton of a convex 3-polytope P, which can be realized in R3 from its skeleton G(P). Various approaches, such as Hart's algorithm and the circle packing theorem, can be used to achieve this and obtain the canonical form of P, which is a polyhedral representation with the highest possible symmetry. In this form, all the edges of P touch a unit sphere. However, instead of an edge-tangent form, it might be desirable to have a circumscribed representation of P where the vertices touch a unit sphere instead of the edges. While not all 3-polytopes are inscribable (e.g., Triakis tetrahedron), can we ensure that for those with inscribability properties, there exists an optimal circumscribed form that preserves the symmetry of the canonical one?