The nine ways of four qubit entanglement and their threetangle

I calculate the mixed threetangle $tau_3[rho]$ for the reduced density matrices of the four-qubit representant states found in Phys. Rev. A {bf 65}, 052112 (2002). In most of the cases, the convex roof is obtained, except for one class, where I provide with a new upper bound, which is assumed to be very close to the convex roof. I compare with results published in Phys. Rev. Lett. {bf 113}, 110501 (2014). Since the method applied there usually results in higher values for the upper bound, in certain cases it can be understood that the convex roof is obtained exactly, namely when the zero-polytope where $tau_3$ vanishes shrinks to a single point.