I've been coming across the condition "IMFCC: having infinitely many finite conjugacy classes" often in recent times and I was wondering if there is any serious difference between having "IMFCC" and having an infinite centre. More precisely:

$\mathbf{Question:}$ Is there a finitely generated group $\Gamma$ which has IMFCC and is not quasi-isometric to a group with infinite centre?

In case the above might be too hard, here is a possible variant: recall a subgroup $M<G$ is almost-malnormal if $\forall g \in G \setminus M,$ the subgroup $gMg^{-1} \cap M$ is finite

$\mathbf{Question~(variant):}$ Let $\Gamma$ be a finitely generated group which has "IMFCC". Is there a subgroup $H < \Gamma$ so that $H$ has infinite centre and if $K$ is another subgroup such that $H < K < \Gamma$ then $K$ is not almost-malnormal?

I do not claim there is any link between quasi-isometry and almost-malnormal subgroups, it's just a geometric and an algebraic condition which seem to express the fact that groups are similar.