operads - Are G_infinity algebras B_infinity? Vice versa?

What is the relationship between $G_infty$ (homotopy Gerstenhaber) and $B_infty$ algebras?

In Getzler & Jones "Operads, homotopy algebra, and iterated integrals for double loop spaces" (a paper I don't well understand) a $B_infty$ algebra is defined to be a graded vector space $V$ together with a dg-bialgebra structure on $BV = oplus_{i geq 0} (V[1])^{otimes i}$, that is a square-zero, degree one coderivation $delta$ of the canonical coalgebra structure (stopping here, we have defined an $A_infty$ algebra) and an associative multiplication $m:BV otimes BV to BV$ that is a morphism of coalgebras and such that $delta$ is a derivation of $m$.

A $G_infty$ algebra is more complicated. The $G_infty$ operad is a dg-operad whose underlying graded operad is free and such that its cohomology is the operad controlling Gerstenhaber algebras. I believe that the operad of chains on the little 2-discs operad is a model for the $G_infty$ operad. Yes?

It is now known (the famous Deligne conjecture) that the Hochschild cochain complex of an associative algebra carries the structure of a $G_infty$ algebra. It also carries the structure of a $B_infty$ algebra. Some articles discuss the $G_infty$ structure while others discuss the $B_infty$ structure. So I wonder: How are these structures related in this case? In general?

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