In some paper, it is written that none of Siegel set of $GL_{n}$ is not contained in any Siegel set of $GL_{n+1}$. I don't understand this because I think it should hold.

To explain my question more precisely, let me first introduce some notations.

Let $G'=GL_n \subset GL_{n+1}$ be general linear groups defined over $\mathbb{Q}$. Let $P_0=U_0M_0$ and $P_0'=U_0'M_0'$ be the standard parabolic subgroups of $G$ and $G'$, respectively. Let $T_0' \subset T_0$ be the maximal split torus of $G'$ and $G$ and $A_0$ and $A_0'$ be the identity component of $T_0(\mathbb{R})$ and $T_0'(\mathbb{R})$.

Let $K \subset K'$ be the standard maximal compact subgroup of $G$ and $G'$. Then by the Iwasawa decomposition, $G=P_0K$ and $G'=P_0'K'$.

Let $\mathfrak{a}_0^*$ be the $\mathbb{R}$-vector space spanned by $\mathbb{Q}$-rational characters of $M_0$ and $\mathfrak{a}_0$ its dual space. Let $\Delta_0$ be the set of simple roots of $\mathfrak{a}_0$.

Let $H:G \mathbb(A_\mathbb{Q}) \to \mathfrak{a}_0$ be the Harish-Chandra function.

For sufficiently negative $t_0 \in \mathfrak{a}_0$, let $A_0(t_0)=\{a \in A_0 \mid \langle \alpha,H(a)-t_0\rangle > 0 \text{ for } \alpha \in \Delta_0\}$.

Then a Siegel set of $G'$ is a set $w'A_0'(t_0')K'$, where $w'$ is a compact subset in $U_0'\mathbb(A_\mathbb{Q})(M_0')^1\mathbb(A_\mathbb{Q})$ and $t_0' \in \mathfrak{a}_0'$. Then for a Siegel set $w'A_0'(t_0')K' \subset G$, can we take a compact subset $w \subset U_0\mathbb(A_\mathbb{Q})(M_0)^1\mathbb(A_\mathbb{Q})$ and $t \in \mathfrak{a}_0$ such that $w'A_0'(t_0')K' \in wA_0(t_0)K$?

I think this is always possible because $P_0' \subset P_0$ and $K' \subset K$. But some paper says that it is not possible. Is there something I am confusing or missing?