Generalized Amit conjecture for fibres of word maps

Let $G$ be a finite nilpotent group, let $w$ be a word in $k$ variables, and define $N_w(g)$ to be the number of tuples $(g_1,\ldots,g_k)\in G^{(k)}$ satisfying $w(g_1,\ldots,g_k)=g$. Let $G_w$ denote the image of the word map $w\colon G^{(k)}\to G$. Generalized Amit conjecture. For every $g\in G_w$, one has

$$N_w(g)\geq |G|^{k-1}.$$

This strengthens Amit's conjecture from the identity fibre to every nonempty fibre of the word map. The source records that the identity case and several special classes of words are known, while the generalized assertion is presented as a version under consideration and remains open.