Densest FM-tetrahedron as a function of the radius ratio

Determine, for each radius ratio r in (0,1) for spheres of radii 1 and r in three-dimensional Euclidean space, whether the densest FM-tetrahedron equals one of five natural extreme configurations (the tight FM-tetrahedra of types 1111, 11rr, and 1rrr; the stretched rrrr tetrahedron with exactly two equal non-tight rr edges and support sphere of radius r; and the stretched 111r tetrahedron with exactly one non-tight 11 edge), as conjectured to hold on specific intervals of r.

Background

In 2D, Florian’s bound identifies a single extremal triangle as densest for disks with radii in [r,1]. In 3D, the authors observe numerically that the densest FM-tetrahedron appears to depend on r, switching among several tight or stretched extremal configurations.

They conjecture a piecewise classification: each of five explicit extremal FM-tetrahedra dominates density over a specific interval of r. Establishing this would be a 3D analogue of Florian’s result, clarifying which local extremal tetrahedra control global density bounds as r varies.