Multilinear Kashiwara–Vergne conjecture

Let $\mathfrak g$ be a finite-dimensional Lie algebra, let $n$ be a positive integer, and let $x_1,\ldots,x_n\in\mathfrak g$. Define

$$\Phi(x_1,\ldots,x_n)=\operatorname{log}(e^{x_1}\cdots e^{x_n}).$$

For each $i$, let $\partial_{x_i}F_i$ be the $\operatorname{End}(\mathfrak g)$-valued real analytic derivative in the $x_i$ direction, and let $\operatorname{tr}$ denote the trace of an endomorphism of $\mathfrak g$. Multilinear Kashiwara–Vergne conjecture. There exist series $F_1,\ldots,F_n$ giving $\mathfrak g$-valued convergent power series on $\mathfrak g^{\times n}$ and satisfying

$$x_1+\cdots+x_n-\operatorname{log}(e^{x_n}\cdots e^{x_1})=(1-e^{-\operatorname{ad}x_1})F_1+(1-e^{\operatorname{ad}x_2})F_2+\cdots+(1-e^{(-1)^n\operatorname{ad}x_n})F_n,$$

and

$$\sum_{i=1}^n\operatorname{tr}(\operatorname{ad}x_i\circ\partial_{x_i}F_i;\mathfrak g)=\frac12\operatorname{tr}\left(\sum_{i=1}^n\frac{\operatorname{ad}x_i}{e^{\operatorname{ad}x_i}-1}+\frac{\operatorname{ad}\Phi(x_n,\ldots,x_1)}{e^{\operatorname{ad}\Phi(x_n,\ldots,x_1)}-1}-1;\mathfrak g\right).$$

The source presents this as the multilinear version of the Kashiwara–Vergne conjecture and explains that its preceding method can be used to find all solutions of the first equation. Its status is not resolved in the supplied text.