Makar-Limanov and Jie-Tai Yu's fractional-power conjecture

Let $K\langle X\rangle$ be a free associative algebra and let ${\mathcal A}(X)$ be the Malcev–Neumann algebra of formal power series with monomials from the free group generated by $X$. Let $g\in K\langle X\rangle$ generate its centralizer, and suppose that the homogeneous component of maximal degree of $g$ is an $n$-th power of an element of $K\langle X\rangle$.

Makar-Limanov and Jie-Tai Yu's conjecture. For every $m>n$ not divisible by $n$, the formal power series

$$g^{m/n}\in{\mathcal A}(X)$$

has a monomial of positive degree containing a negative power of a generator. This conjecture was formulated in the authors' approach to degree estimates for commutators and would imply Jie-Tai Yu's degree estimate conjecture. Its status is not resolved in the supplied text.