Balanced infinitesimal bialgebras, double Poisson gebras and pre-Calabi-Yau algebras

We consider the properad that governs the balanced infinitesimal bialgebras equipped with a coproduct of degree $1-d$. This properad naturally encodes a part of the structure of the pre-Calabi-Yau algebras of degree $d$. We compute the cobar construction of its Koszul dual coproperad and show that its gebras lie between the homotopy double Poisson gebras and the pre-Calabi-Yau algebras. Finally, we show that, if one is willing to consider their curved version, the two resulting notions of curved homotopy balanced infinitesimal bialgebra and curved homotopy double Poisson gebra are equivalent. A relationship with the homotopy odd Lie bialgebras is also discussed.