This is a follow-up to Does the Bruhat decomposition induces decomposition on integral points (on an open cell)?

Given a split connected reductive group $G$ over $\mathbb Q_p$, suppose $N_+$ is the unipotent radical of a Borel and $N_-$ its opposite. Is it true that $$G(\mathcal O) \cap N_-(F)N_+(F) = N_-(\mathcal O)N_+(\mathcal O)?$$ I can prove this for $G=\mathrm {GL}_n$ basically by a sort of induction on the rows and columns in the matrix, so I'd be happy with a reduction to this case. But I'm also curious if there are ways to prove this that don't involve reducing to $\mathrm{GL}_n$.

This is a follow-up to Does the Bruhat decomposition induces decomposition on integral points (on an open cell)?

Given a split connected reductive group $G$ over a $p$-adic local field $F$ with ring of integers $\mathcal O$, suppose $N_+$ is the unipotent radical of a Borel and $N_-$ its opposite. Is it true that $$G(\mathcal O) \cap N_-(F)N_+(F) = N_-(\mathcal O)N_+(\mathcal O)?$$ I can prove this for $G=\mathrm {GL}_n$ basically by a sort of induction on the rows and columns in the matrix, so I'd be happy with a reduction to this case. But I'm also curious if there are ways to prove this that don't involve reducing to $\mathrm{GL}_n$.