Exact critical threshold for simple games

Let $c_\mathcal{S}(n)$ denote the maximum critical threshold value among simple games on $n$ voters. Simple-game threshold conjecture. For every $n\ge 4$,

$$c_\mathcal{S}(n)=\frac{\left\lfloor\frac{n^2}{4}\right\rfloor}{n}.$$

The paper proves the displayed quantity as a lower bound, improving the previously known bound for odd $n$, but does not establish the matching upper bound.