Boros–Füredi–Kelly conjecture on skew lines in four-dimensional space

Let $\Sigma$ be a finite family of pairwise skew lines in $\mathbb{R}^4$. Suppose that the three-dimensional affine span (hull) containing any two of the lines contains at least two more lines of $\Sigma$. Boros–Füredi–Kelly conjecture. Then the family is contained in a single three-dimensional space. The paper presents this as a further conjecture related to an elementary proof of Serre's question; the supplied text gives no evidence resolving it.