Here is a brief hint: Each list of l-group terms represents an arrow diagram as described by Holland and McCleary. The empty string is the starting point of the diagram, and the number following the list gives the position of the endpoint. If the position of the endpoint is >= the position of the empty string, for all diagrams, then the terms is >= e. For example:

[["xyXY", "", "xyX", "x", "xy"], 0] shows that xyx^{-1}y^{-1}>=e fails in l-groups since the term in position 0 of the list is to the left of "". (Many other diagrams show this as well.)