Shapley–Shubik one-third approximation conjecture

Let $n\ge3$, let $\sigma\in\mathbb{R}_{\ge0}^n$ satisfy $\Vert\sigma\Vert_1=1$, and write $\sigma^n$ for the desired power distribution in dimension $n$. Shapley–Shubik approximation conjecture. There exists a weighted majority game $v$ such that

$$\left\Vert\operatorname{SSI}(v)-\sigma^n\right\Vert_1\le\frac13.$$

This asks for a uniform approximation guarantee over the unit simplex; its resolution is not given in the supplied text.