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".

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).

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.

Maybe RAGlib can do a better job here.

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.

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".