  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.
