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We give an exact criterion of a conjecture of L.M.Kelly to hold true which isstated as follows. If there is a finite family $\Sigma$ of mutually skew linesin $\mathbb{R}^l,l\geq 4$ such that the three dimensional affine span (hull) ofevery two lines in $\Sigma$, contains at least one more line of $\Sigma$, thenwe have that $\Sigma$ is entirely contained in a three dimensional space if andonly if the arrangement of affine hulls is central. Finally, this article leadsto an analogous question for higher dimensional skew affine spaces, that is,for $(2,5)$-representations of sylvester-gallai designs in $\mathbb{R}^6$,which is answered in the last section.

On a Conjecture of Kelly on $(1,3)$-representation of Sylvester Gallai Designs