Linear groups saturated by subgroups of finite central dimension

Let $F$ be a field, $A$ be a vector space over $F$ and $G$ be a subgroup of $\mathrm{GL}(F,A)$. We say that $G$ has a dense family of subgroups, having finite central dimension, if for every pair of subgroups $H$, $K$ of $G$ such that $H\leqslant K$ and $H$ is not maximal in $K$ there exists a subgroup $L$ of finite central dimension such that $H\leqslant L\leqslant K$. In this paper we study some locally soluble linear groups with a dense family of subgroups, having finite central dimension.