In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of this theorem, which I don't know if already exists (or even if it's true).

My starting operad $\mathcal{P}$ is a dg-operad (note that, in the version of the HTT I mentioned, the starting operad does not have a differential), not necessarily Koszul (if this makes sense). Let $A$ be a dg-$\mathcal{P}$-algebra, and $i: H \to A$, $p: A \to H$, $h: H \to H$ a homotopy retract of $A$. Is it true that $H$ has a $\Omega B \mathcal{P}$-algebra structure such that $i: H \to A$ is a quasi-isomorphism (or leads to a zig-zag of quasi-isomorphisms)?

*Note:* Here $\Omega B \mathcal{P}$ denotes the bar-cobar construction of the dg-operad $\mathcal{P}$.

My thoughts on this are that it is true, and that the formulas given in the book of Loday and Vallette for the construction of the homotopy algebra on $H$ could be extended in this context. However, I would like to know if this is already proven (so I can avoid writing a proof with a horrendous notation), or if this is known to be false (in which case I would love to see a counterexemple!).

Thanks in advanced :)