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.

Here is this [calculation at SageCell](https://sagecell.sagemath.org/?z=eJxdkctuwjAQRfdI_MPsYrduC0GqKiSzYV-JsqQBuY6DTPwIeSAjxL93HB5F9W7mXp25nslVAfbYeHNQhE6HA8A3Tz-BwwnarjKKSDotofA1lEyCdqBcZ1UtWkXm3v5oJ1rtXUPeWUopnIeDCyMgofLmuFWoLRbMKEeQS1kSEnozFWhqOksgrFBbEc12NMue7lX5UO36qg-CNlbGKP8CTCjc0RLRB1GTRMZxsbOP0_aqkiLfIMV2RrwKl28IrJZfJGg646MeH3RkhwyeeyUGDHqdPmqUcj7uDRIFzgFjeKuF6VNg6-2xk9L15EYrKMxAZnANVdXataRIvtslnkC7LZz2xflvQ7Vqu9pdYxP8AwOrrFTGNHw8Wn8wwNN1cQE8Ee74UnnkJZc13M_6C1t-jCo=&lang=sage&interacts=eJyLjgUAARUAuQ==), which takes about a minute or so and results in an error saying *"input formula is false everywhere"*.