Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation.

Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular semisimple)) such that there exists $g\in GL_{n}(O)$ such that

$gAg^{-1}=B$,

Does there exist an element $g_{1}\in G(O)$ such that $g_{1}Ag_{1}^{-1}=B$?

Let $k$ an algebraically closed field. Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.

Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular semisimple)) such that there exists $g\in GL_{n}(O)$ such that

$gAg^{-1}=B$,

Does there exist an element $g_{1}\in G(O)$ such that $g_{1}Ag_{1}^{-1}=B$?

Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation.

Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular semisimple)) such that there exists $g\in GL_{n}(O)$ such that

$gAg^{-1}=B$,

does it exist an element $g_{1}\in G(O)$ such that $g_{1}Ag_{1}^{-1}=B$?

Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation.

Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular semisimple)) such that there exists $g\in GL_{n}(O)$ such that

$gAg^{-1}=B$,

Does there exist an element $g_{1}\in G(O)$ such that $g_{1}Ag_{1}^{-1}=B$?