Geproci sets on skew lines in P^3 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 ≤ 6. Moreover we show that all geproci sets of this type and with no points on the transversals are contained in the F4 configuration. We conjecture that a similar result is true for an arbitrary number m of points on each skew line, replacing containment in F4 by containment in a half grid obtained by the so-called standard construction.