Sharp nucleolus norm bound for normalized weights

Let $[q;w_1,\dots,w_n]$ be a weighted majority game with $q\in(0,1)$ and $w\in\mathbb{R}_{\ge0}^n$ normalized by $\Vert w\Vert_1=1$. Let $\Delta=\max_{1\le i\le n}w_i$, and let $\operatorname{Nuc}$ denote the nucleolus. Nucleolus norm-bound conjecture.

$$\left\Vert \operatorname{Nuc}([q;w_1,\dots,w_n])-(w_1,\dots,w_n)\right\Vert_1\le\frac{\Delta}{\min(q,1-q)}$$

is valid and tight for normalized weights. The surrounding discussion explains that such bounds control the error obtained by using desired power distributions as weights, but the stated sharp bound is not resolved in the supplied text.