Non-Koszulness in a family of properads

Proving Koszulness of a properad can be very hard, but sometimes one can look at its Koszul complex to look for obstructions for Koszulness. In this paper, we present a method and tools to prove non-Koszulness of many properads in a family of quadratic properads. We illustrate this method on a family of associative and coassociative properads with one quadratic compatibility relation.