Ginzburg--Kapranov criterion for Koszulness of operads, and its limitations

In their seminal paper on operadic Koszul duality, Ginzburg
and Kapranov established a remarkable functional equation that holds
whenever an operad is Koszul. I shall discuss some examples
demonstrating that non-Koszul operads do not have to violate that
criterion, in either of two possible ("weak" or "strong") senses.