Ziegler's conjecture on power bases of units in pure number-field orders

Let $d\geq 2$ be an integer and let $m\in\mathbb{Z}\setminus\{0\}$ be such that $\sqrt[d]{m}$ has algebraic degree $d$. The order $\mathbb{Z}[\sqrt[d]{m}]$ is the subring generated by this algebraic integer, and a power basis consisting of units is a basis of the form $1,\alpha,\alpha^2,\ldots,\alpha^{d-1}$ whose elements are all units for a suitable generator $\alpha$. Ziegler's conjecture. The order $\mathbb{Z}[\sqrt[d]{m}]$ admits a power basis consisting of units if and only if $m=a^d\pm1$ for some integer $a$. This extends the known quartic result and analogous results in lower degrees; the source gives no resolution of the general statement.