This problem can be posed as finding a point satisfying a system of polynomial equations and inequalities. There exist a few software for solving them, such as [QEPCAD](https://www.usna.edu/CS/qepcadweb/B/QEPCAD.html) (available in Sage) and [RAGLib](https://www-polsys.lip6.fr/~safey/RAGLib/) (available in Maple).

*UPDATED*. I've supplied the system composed of $\sum_{i<j} x_{i,j}^2=1$, $x_{i,j}\geq 0$, $c^2 = \binom{6}{3}^2/\binom{6}{2}^3$, $c>0$, and $\sum_{i<j<k} x_{i,j}x_{i,k}x_{j,k} > c$ to QEPCAD, requesting to find any point satisfying it, and it said that such point does not exist. However, along the way it comes with a lot of warnings about *"The McCallum projection may not be valid."*
Here is this [calculation at SageCell](https://sagecell.sagemath.org/?z=eJxdkc1uwjAQhO9IvMPebLdugSBVFZK5cK9EOdKAXMdBJv4JiYOCEO_eTaAU1SfvzujzeDfTObhTHexRUzYbDgDPIvkAAWeITWk1VWxWQB4qKLgC40H7xulKRk0XwX0bL6MJvqZvPGEMLsPBldEioQz2tNOoLZfcak-RyzhpCfs15WiqG0ehXaO2pobvWZo-3aviodr3VR8EbbzoovwLMGVwRytEH2VFieqe6zqH7rWDLpXMtkhxjZWv0mdbCuvVJ20Nm4txj29Nx25TeO6VLmBrNsmjxpgQk96gUBACMEZwRto-BbZGj52EbaYcE81hzAGJOcOrSuEWrKyMjzQnX3GFazB-B-dDfvmbUqVjU_lbdIr_4OC0U9raWkzGm3cOuL6mG4Ig0p9eyoA8wsGGXW6sFmSU6eOojlloIrlO6L7xH1uqlKk=&lang=sage&interacts=eJyLjgUAARUAuQ==).

Maybe RAGlib can do a better job here.