Negative explicit-bound conjecture for the Public Good Index

Let $P$ be a power index, let $[q;w_1,\dots,w_n]$ denote a weighted majority game with $q\in(0,1)$, normalized nonnegative weights $w\in\mathbb{R}_{\ge0}^n$ satisfying $\Vert w\Vert_1=1$, and let $\Delta=\max_{1\le i\le n}w_i$. Public Good Index bound conjecture. No constants $\alpha,\beta,c\in\mathbb{R}_{>0}$ make

$$\left\Vert P([q;w_1,\dots,w_n])-(w_1,\dots,w_n)\right\Vert_1\le\frac{c\cdot\Delta^\alpha}{\min(q,1-q)^\beta}$$

hold for all $n\in\mathbb{N}$, $q\in(0,1)$ and such normalized weight vectors when $P$ is the Public Good Index. This conjecture asserts that the explicit-bound question has a negative answer for that index; the source gives no resolution.