Let $G$ a locally profinite not compact group and we suppose that $G$ have maximal compact subgroups, for example $G$ reductive group over p-adic field.
We call projective Tower of $G$ any strictly decreasing sequence $\mathcal{K}=(K_{n})_{n\in\mathbb{N}}$ of open compact subgroup of $G$. We define an ordering on the set $\mathcal{Pt}(G)$ of projective tower of $G$ as follows , $\mathcal{K}\leqslant\mathcal{L}$ if and only if $\mathcal{K}$ is a sub-sequence of $\mathcal{L}$, it is not difficut to verify that is an ordering.
We call maximal projective tower of $G$ any $\mathcal{K}\in\mathcal{Pt}(G) $ maximal for this ordering. It is not difficut to verify that there exists such elements and for such element $\mathcal{K}=(K_{n})_{n\in\mathbb{N}}$ the first term of the sequence is maximal compact subgroup of $G$.
The group $G$ acts on the set $\mathcal{Pt}_{max}(G)$ of maximal projective tower of $G$ by conjugation.
My question is : Is this action are transitive ? I think is not transitive.
