In reduction theory of arithmetic groups, one has the following finiteness property.

Proposition 1 (Siegel property).Let $G$ be a reductive group over $\mathbb{Q}$ and let $\Gamma\subset G(\mathbb{Q})$ be an arithmetic subgroup. Let $\mathfrak{S}\subset G(\mathbb{R})$ be a Siegel set. Then the set $$\{\gamma\in \Gamma; \mathfrak{S}\cap \gamma. \mathfrak{S}\ne \emptyset\}$$ is finite.

Now I am consider a generalization of the above finiteness property as following.

Question 2Let $G$ be a reductive group over $\mathbb{Q}$ and let $\Gamma\subset G(\mathbb{Q})$ be an arithmetic subgroup. We fix $\mathfrak{S}\subset G(\mathbb{R})$ be a Siegel set. Take $H\subset G$ be a reductive subgroup over $\mathbb{Q}$. Then is there a Siegel set $\mathfrak{S}_H\subset H(\mathbb{R})$ such that the set $$\{\gamma\in \Gamma; \mathfrak{S}\cap \gamma. \mathfrak{S}_H\ne \emptyset\}$$ is finite?

So is this generalization still true? Could you please provide a proof or a counter example? Thanks very much!