If $G$ is a group with finite Hirsch number and with its maximal locally finite normal subgroup satisfying the minimal condition on subgroups, e.g. if $G$ is a finite extension of a torsion-free soluble group of finite rank, then there exists an integer $k=k(G)$ such that for every subgroup $H$ of $G$ any chain of idempotents in the endomorphism monoid $\mathrm{End}(H)$ of $H$ has length at most $k$.