Let $G$ be arbitrary group. Let us assume it is $\operatorname{FP}_\infty$. Suppose that the integral cohomology groups $H^i(G, \mathbb{Z})$ have bounded rank as finitely generated free abelian groups for all $i \geq 0$, i.e. exists $N \geq 0$ such that $\mathrm{rk}_{\mathbb{Z}}H_i(G, \mathbb{Z}) \leq N$ for every $i \geq 0$. Is there name for this property? Of course each $H^i(G, \mathbb{Z})$ is finitely generated (but not necessarily with upper bound on their ranks across all $i$) as $G$ is $\operatorname{FP}_\infty$ but I am interested in this (apparantly) stronger property. Is it equivalent to $\operatorname{FP}_\infty$? Is there a group of type $\operatorname{FP}_\infty$ which is not of this type? I am equally interested in the homology case $H_i(G, \mathbb{Z})$.
Obviously free groups have this stronger property as do torsion-free hyperbolic groups, and any group of type $\operatorname{FP}$.
