Banzhaf lower bound for the two-level target distribution

For each $n\ge2$, define

$$\psi^n=\frac{1}{2n-1}(2,\dots,2,1)\in\mathbb{R}_{\ge0}^n.$$

A weighted majority game is a game represented by a quota and nonnegative voter weights. Two-level Banzhaf lower-bound conjecture. There is a constant $c>0$ such that, for every $n\in\mathbb{N}_{\ge2}$ and every weighted majority game $v$,

$$\left\Vert\operatorname{Bz}(v)-\psi^n\right\Vert_1\ge\frac{c}{n}.$$

The bound would show an unavoidable order-$1/n$ error for this oceanic sequence of targets; the supplied text does not resolve it.