Flip distance and triangulations of a polyhedron

It is known that the flip distance between two triangulations of a convex polygon is related to the smallest number of tetrahedra in the triangulation of some polyhedron. The latter was used to compute the diameter of the flip graph of convex polygons with a large number of vertices. However, it is yet unknown whether the flip distance and this smallest number of tetrahedra are always the same or even close. In this work, we find examples to show that the ratio between these two numbers can be arbitrarily close to $\frac{3}{2}$. We also propose two conjectures in the end, one about this ratio, and the other may have some implications on when two triangulations can achieve maximal distance.