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).

I've supplied the system composed of $\sum_{i<j} x_{i,j}^2=1$, $x_{i,j}\geq 0$, $c^2 = \binom{6}{3}^2/\binom{6}{2}^3$, 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.