Let $\Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in { \rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $\mathfrak{ g}(\Sigma)$ and its associated graded ${ \rm gr}\, \mathfrak{ g}(\Sigma)$. In this paper, we prove the converse: if $F$ induces an isomorphism $\mathfrak{ g}(\Sigma) \cong { \rm gr} \, \mathfrak{ g}(\Sigma)$, then it satisfies the Kashiwara-Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov-Kostant-Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.
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