Theta-square-root growth of complete-game critical thresholds

Let $c_\mathcal{C}(n)$ denote the maximum critical threshold value of a complete simple game on $n$ voters. Complete-game growth conjecture.

$$c_\mathcal{C}(n)\in\Theta\!\left(\sqrt{n}\right).$$

This is the asymptotic form of the paper's conjectured upper and lower bounds. The lower bound is achieved by an explicit class of examples, while the general upper bound remains open.