Suppose $W$ is a cyclic $L_\infty$ algebra, i.e. $W$ has a non-degenerate, symmetric, invariant pairing $\langle\cdot,\cdot\rangle_W$. Let $V$ be a cochain complex, and suppose given the data of a strong deformation retraction $(i,p,k)$ of $W$ onto $V$, i.e. $p$ is a cochain map $W\to V$, $i$ is a cochain map $V\to W$, $pi$ is the identity on $V$, $k$ is a homotopy between $ip$ and the identity on $W$, and we have the following three "side conditions:" $k^2=ki=pk=0$.
Given these data, the homotopy transfer theorem guarantees that $V$ has the structure of $L_\infty$ algebra such that $i$ extends to an $L_\infty$ map. Define $\langle v_1,v_2\rangle_V:= \langle i(v_1),i(v_2)\rangle_W$, and suppose that $\langle\cdot,\cdot\rangle_V$ is non-degenerate. Under what conditions on the data $(i,p,k)$ is $\langle\cdot,\cdot\rangle_V$ invariant for the $L_\infty$ structure on $V$ given by homotopy transfer?
I have encountered the notion of "cyclic" strong deformation retraction, in which one requires $\ker(p)\perp \text{im}(i)$ and $\langle kw_1,w_2\rangle_W = \langle w_1,kw_2\rangle_W$. Are these sufficient conditions for $V$ (with the induced operations) to be a cyclic $L_\infty$-algebra?
