The Amit–Ashurst conjecture for finite nilpotent groups

Let $w$ be a word in $k$ variables, let $G$ be a finite group, and let $G_w$ denote the set of values of $w$ in $G$. For $gin G$, write

$$P_{w,G}(g)=\frac{\left|\{(g_1,\ldots,g_k)\in G^k\mid w(g_1,\ldots,g_k)=g\}\right|}{|G|^k}.$$

Amit–Ashurst conjecture. Let $G$ be a finite nilpotent group. Then

$$P_{w,G}(g)\geq |G|^{-1},$$

where $w$ ranges over all words and $g\in G_w$. This strengthens the corresponding lower-bound property known for abelian, nilpotent dihedral, and generalized quaternion groups; the paper proves the conjecture for finite $p$-groups with a cyclic maximal subgroup, while the general finite nilpotent case remains open.