I understand,that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with $N$ tetrahedra.

Is it known how many equilateral tetrahedra we can fit inside a triangulated sphere with boundary $A$ (in Euclidean space) ?

Is it known how many triangulations are there inside a sphere with boundary $A$?