Asymptotic bounds for the Nakamura number of weighted games

Let $\operatorname{Nak}^{\mathcal{T}}(n,t)$ denote the maximum possible Nakamura number among weighted games on $n$ players with $t$ types of players, where $t\in\mathbb{N}_{>0}$. If $n$ is sufficiently large, then the asymptotic Nakamura-number conjecture.

$$n-t+1\le \operatorname{Nak}^{\mathcal{T}}(n,t)\le n-t+2.$$

The preceding proposition establishes the corresponding values for $t\le 4$, while the paper leaves the determination of the three Nakamura-number functions for $t>4$ open. Thus the stated bounds are supported by small-parameter computations but remain open in general.