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 (available in Sage) and 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.
Here is this calculation at SageCell, which takes about a minute or so and results in an error saying "input formula is false everywhere".