We say that the ${{\bf Ger}}^+_{\infty}$-algebra structures ${\hat{Q}}$ and ${\hat{Q}}'$ are homotopic if there exists a degree $0$ coderivation ${\hat{\psi}}$ of the coalgebra ${{\bf S}}(V,A)$ satisfying $$\label{conjugated}
{\hat{Q}}' = \exp([{\hat{\psi}}, \,]) {\hat{Q}}\,.$$

\[rem:homot-q-iso\] Homotopic ${{\bf Ger}}^+_{\infty}$-structures on a pair $(V,A)$ are obviously quasi-isomorphic. Namely, a ${{\bf Ger}}^+_{\infty}$ quasi-isomorphism from $(V,A, {\hat{Q}})$ to $(V,A, {\hat{Q}}')$ is given by the ${{\bf S}}$-coalgebra automorphism $\exp({\hat{\psi}})$ of ${{\bf S}}(V,A)$.

Definition \[dfn:homotopies\] is justified by Proposition 4.10 from paper[^4] [@ChuangLazarev] by J. Chuang and A. Lazarev. Indeed, let ${{\rm E n d}}_{(V,A)}$ be the endomorphism operad of the pair $(V,A)$. Then Proposition 4.10 from [@ChuangLazarev] implies that ${{\bf Ger}}^+_{\infty}$-algebra structures ${\hat{Q}}$ and ${\hat{Q}}'$ are in the relation (\[conjugated\]) if and only if the corresponding maps of DG operads $${{\rm Cobar}}({{\bf S}}) \to  {{\rm E n d}}_{(V,A)}$$ are homotopic with the homotopy defined via a path object.

We will be interested in homotopies between ${{\bf Ger}}^+_{\infty}$-algebra structures for which the coderivation $${\hat{\psi}}\in {{\rm Coder}}({{\bf S}}(V,A))$$ satisfies the following additional conditions $$\label{condition-V}
{\hat{\psi}}\Big|_{{{\bf Ger}}^{\vee}(V)} = 0\,.$$ $$\label{condition-A}
{\hat{\psi}}\Big|_{ {{\underline{T}}}({{\bf s}}^{-1}A)} = 0\,.$$ These conditions are motivated by the following obvious proposition:

\[prop:conditions\] Let $(V,A)$ be a pair of cochain complexes and ${\hat{Q}}$, ${\hat{Q}}'$ be ${{\bf Ger}}^+_{\infty}$-algebra structures 