The relative Banzhaf index upper-bound conjecture

Let $w_1,w_2,w_3, \ldots, w_m$ be positive weights, let $w$ be their maximum, and set

$$N=\sum_{i=1}^{m}w_i.$$

For a voting game with these weights, let $\beta_i$ denote the relative Banzhaf index of player $i$. Relative Banzhaf index upper-bound conjecture. For every player $i$,

$$\beta_i\leq\frac{2w}{N}.$$

The authors report that this relation was observed in their calculations for voting games but had not been proved. It is presented as a narrow bound on the relative Banzhaf indices.