Geproci sets on skew lines in P3 with two transversals

The purpose of this work is to pursue classification of geproci sets. Specifically we classify $[m,n]$-geproci sets Z which consist of $m=4$ points on each of n skew lines, assuming the skew lines have two transversals in common. We show in this case that $n\le 6$. Moreover we show that all geproci sets of this type and with no points on the transversals are contained in the $F_4$ configuration. We conjecture that a similar result is true for an arbitrary number m of points on each skew line, replacing containment in $F_4$ by containment in a half grid obtained by the so-called standard construction.