Strict monotonicity conjecture for Maker–Breaker clique games

For positive integers $m,b$ and a positive integer $N$, let $f_N(m,b)$ be the largest integer $q$ such that Maker can occupy a clique $K_q$ in the $(m:b)$ Maker–Breaker game played on $K_N$, where Maker claims $m$ edges and Breaker claims $b$ edges per turn.

Strict monotonicity conjecture. For fixed $b,m$, for all sufficiently large $N$,

$$f_N(m,b+1)<f_N(m,b)<f_N(m+1,b),$$

and both differences tend to infinity as $N\to\infty$.

The source notes that the first inequality for $m=1$ would follow from an unsolved conjecture of Bednarska and Łuczak. The general conjecture is therefore presented as open.