Clark and Ismail's positivity conjecture for derivatives of a quotient

Let $n\in\mathbb{N}$ and define the function $x\mapsto x^n/(1-e^{-x})$ on $(0,\infty)$. Clark and Ismail's positivity conjecture. For all $x\in(0,\infty)$ and $n\in\mathbb{N}$,

$$\left(\frac{x^n}{1-e^{-x}}\right)^{(n)}>0.$$

This conjecture was disproved: Alzer, Berg, and Koumandos proved that for all sufficiently large integers $m$, the related function $G_m^{(m)}$ is not completely monotonic, and the displayed Laplace-transform relation implies that the asserted positivity fails for some $x$ and $n$.