Equivalence between the noncommutative krv1 equation and coefficient identities

Let $k$ be the coefficient field, let $\mathfrak{fr}_k(x_0,x_1)$ be the completed free Lie algebra on $x_0,x_1$, and let $\psi_{451}$, $\psi_{123}$, and $l^{y,x}_{(a_1,\dots,a_m),(b_1,\dots,b_n)}$ have the meanings defined in the paper. Assume that $\psi\in\mathfrak{fr}_k(x_0,x_1)$ is skew-symmetric and satisfies $c_{x_0}(\psi)=c_{x_1}(\psi)=0$. For sequences $(a_1,\dots,a_m)$ and $(b_1,\dots,b_n)$, exclude the pair in which both sequences are $(1,\dots,1)$. The noncommutative krv1 equivalence conjecture. The following conditions are equivalent: (i) $\psi$ satisfies the krv1 equation

$$[x_1,\psi(-x_0-x_1,x_1)]+[x_0,\psi(-x_0-x_1,x_0)]=0;$$

(ii) for every admissible pair of sequences,

$$l^{y,x}_{(a_1,\dots,a_m),(b_1,\dots,b_n)}(\psi_{451}+\psi_{123})=l^{y,x}_{(a_1,\dots,a_m,b_1),(b_2,\dots,b_n)}(\psi_{451}+\psi_{123}).$$

This conjecture seeks to characterize the noncommutative krv1 equation through the stated coefficient identities, within the skew-symmetric part of the completed free Lie algebra. The notation $\psi_{451}$, $\psi_{123}$, and the coefficient functionals $l^{y,x}$ must be interpreted from the paper's preceding definitions; the source does not provide resolution evidence for the conjecture.