Let $G$ be a discrete group and $X$ be a $G$-CW complex. For any $x\in X$ and open neighborhood $U$ of $x$, I am interested in the question that whether we can find a $G_x$-invariant open neighborhood $V$ of $x$ that is contained in $U$. This is not true in general, for example, if $X$ is the cone over $\mathbb R$ with the $G=\mathbb Z$ action by translation, then the cone point $x_0$ has a neighborhood which does not contain any $G_{x_0}$-invariant open neighborhood. But I am wondering whether the following is true:

Any $G$-CW complex $X$ is $G$-homotopy equivalent to a $G$-CW complex $Y$ so that any open neighborhood $U$ of $y\in Y$ contains a $G_y$-invariant open neighborhood $V$.

As a special case, I am wondering whether the following is true:

Let $\mathcal F$ be a family of subgroups of $G$ which is closed under conjugation and finite intersection. Can we always find a model $E_{\mathcal F}G$ for the universal $G$-CW complex relative to $\mathcal F$ so that any open neighborhood of $x\in E_{\mathcal F}G$ contains a $G_x$-invariant open neighborhood?

Thank you!