Aichholzer's compatible triangulation conjecture with prescribed convex-hull bijection

Let $P$ and $Q$ be two arbitrary point sets in general position with $|P|=|Q|$. Write $\operatorname{CH}(P)$ and $\operatorname{CH}(Q)$ for their convex hull vertex sets, and let a triangulation be a maximal set of pairwise non-intersecting segments joining pairs of points. A pair of triangulations $(T_P,T_Q)$ is compatible with respect to a bijection $f:P\to Q$ if $(p_i,p_j)\in T_P$ exactly when $(f(p_i),f(p_j))\in T_Q$. Aichholzer's compatible triangulation conjecture. For any bijection $f_0:\operatorname{CH}(P)\to\operatorname{CH}(Q)$ preserving the cyclic order of the points on the convex hull, there exists a bijection $f:P\to Q$ extending $f_0$ such that $(P,Q)$ has a compatible triangulation with respect to $f$. The paper proves special cases, including the double-circle order type and some generalizations, while the full assertion remains open.