Arbitrary-field fractional-power conjecture

Let $K$ be an arbitrary field and let $g\in K\langle X\rangle$ generate its own centralizer. Consider a pair $(m,n)$ with $m,n\in\mathbb{N}$, $m,n\geq 2$, and $n\nmid m$. Assume that $g^{1/n}$ makes sense in $K((X))$, meaning that there is some $h\in K((X))$ such that

$$g=h^n.$$

Arbitrary-field fractional-power conjecture. Then $g^{m/n}\in K((X))$ has a monomial of positive degree containing a negative power of an indeterminate in $X$.

This modifies the characteristic-zero fractional-power conjecture so that it can be stated over an arbitrary field, and the paper says it is equivalent to the characteristic-zero version when the characteristic is zero.