Amit's conjecture on word-map fibers in finite nilpotent groups

Let $G$ be a finite nilpotent group, let $\overline{x}$ denote a tuple of variables, and let $w(\overline{x})$ be a word map on $G$. Write $\mu_{G,w}(1)$ for the probability that $w(\overline{x})=1$ when $\overline{x}$ is chosen uniformly from $G^{|\overline{x}|}$. Amit's conjecture. For every such word map,

$$\mu_{G,w}(1)\geq \frac{1}{|G|},$$

i.e. the equation $w(\overline{x})=1$ has at least $|G|^{|\overline{x}|-1}$ solutions. This generalizes the preceding result for equations of the form $w(\overline{x})=c(\overline{x})$ with $c$ in the derived subgroup of the free group over $G$; its status is not resolved in the supplied source context.