Monotonicity conjecture for the relative Banzhaf power index

Let $n_1,n_2$ be the numbers of players of the two types, and let $Bz_2(a,b,n_1,n_2)$ denote the relative Banzhaf power index of a player of type $2$ in the corresponding complete simple game. Relative Banzhaf monotonicity conjecture.

For $b\geq 1$,

$$Bz_2(a,b,n_1,n_2)-Bz_2(a,b-1,n_1,n_2)>0.$$

For $a\geq 2$,

$$Bz_2(a-1,b,n_1,n_2)-Bz_2(a,b,n_1,n_2)\geq 0,$$

with equality for $b=0$ and a strict inequality otherwise. The conjecture would imply a corollary analogous to the stated ordering result for the relative Banzhaf index; it is supported in the source by verification for all $n\leq 100$.