Has anyone defined what it means for a modular operad to be Koszul, or what the Koszul dual of a modular operad is? In particular, is it meaningful to say that a modular operad is quadratic? Merkulov, Markl and Shadrin (Wheeled PROPs, graph complexes and the master equation) give a definition for wheeled properads, which are basically modular operads with oriented edges, which are constrained to have at most one output.

Presumable the role of bar-cobar duality is then played by the Feynman transform and the dual Feynman transform.