Let $G$ be an almost-simple classical real algebraic group, and $H$ a maximal proper closed subgroup. Assume that the $n$ in $G := A_n,B_n,C_n,D_n$ is sufficiently large.

Is it true that dim$(G)$ - dim$(H)$ $\geq$ O(rank($H$))? where $O$ stands for the big-O notation.

Let $G$ be an almost-simple classical real algebraic group, and $H$ a maximal proper closed subgroup. Assume that the $n$ in $G := A_n,B_n,C_n,D_n$ is sufficiently large.

Is it true that dim$(G)$ - dim$(H)$ $\geq$ rank($H$)?