Cyclic-word presentation conjecture for the free associative Lie algebra

Let $A_n$ be the free associative algebra on generators $x_1,\dots,x_n$, regarded as a Lie algebra with bracket $[a,b]=a\cdot b-b\cdot a$. For a monomial $a=x_{i_1}\cdots x_{i_\ell}$, let

$$\widetilde a=x_{i_1}\cdots x_{i_\ell}+x_{i_2}\cdots x_{i_\ell}x_{i_1}+\cdots+x_{i_\ell}x_{i_1}\cdots x_{i_{\ell-1}}$$

be its cyclic sum, and let $R(a,b,c)$ be the cyclic relation obtained in the source from the associative identity $[a,b\cdot c]+[b,c\cdot a]+[c,a\cdot b]=0$. Cyclic-word presentation conjecture. The Lie algebra $A_n$ is generated by the cyclic words $\widetilde a$ for monomials $a\in A_n$, with relations $R(a,b,c)$ for every triple of monomials $a,b,c\in A_n$.