Eliminating the obstructive d=4 higher-dimensional configuration

Prove that in the d=4 case with j=2, the triples from a common 3-flat associated with a given pair cannot be almost all contained in a line, or otherwise establish a construction guaranteeing conditions (i) and (ii) of balanced supersaturation simultaneously; extend the result to the analogous obstruction in all higher dimensions.

Background

The proof of balanced supersaturation in dimension three divides according to the structure supplied by Claims 6 and 7. The authors state that analogous arguments handle the cases j=3 and j=4 when d=4, but the case j=2 requires adding three points from U in the same 3-flat as a given pair.

The difficulty is that almost all such triples might be contained in a line, preventing the authors from ensuring both the required number of configurations and the required codegree bounds. The authors believe this obstruction should not occur, but explicitly state that they do not know how to prove this; the same barrier is said to persist in all higher dimensions.