Conjecture that the Sommerville and Goldberg lists classify face-to-face tetrahedral tiles

A face-to-face tetrahedral tile is a tetrahedron that tiles space with tile intersections occurring face-to-face. The Sommerville and Goldberg lists are the previously known lists of such tiles.

Sommerville–Goldberg classification conjecture. The Sommerville and Goldberg lists of face-to-face tetrahedral tiles are complete. In particular, every tetrahedral tile has at least one dihedral angle equal to $\pi/2$.

This would complete the classification of face-to-face tetrahedral tiles. The paper states it as an unresolved conjecture.