I have difficulties finding an appropriate reference for the following question (which I hope that it to be true).

Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexiﬁcation and $\tau: U^{\mathbb{C}} \rightarrow GL(V)$ a holomorphic representation of $U^{\mathbb{C}}$ on a complex vector space $V$. Let $v,w \in V$ such that $w$ lies in the boundary of the orbit of $v$ ($w \in \partial(G\cdot v)$). The question is:

Is there a open neighbourhood $\triangle_{w}$ of $w$ such that there are a $v\prime \in G\cdot v \cap \triangle_{w}$ and $X \in \mathfrak{g}_{w}$ such that

$$\lim_{t \to \infty} \exp(tX)\cdot v\prime = w?$$

Here, $\mathfrak{g}_{w} :=\{X \in \operatorname{Lie}(G) : X\cdot w=0\}$ where $\operatorname{Lie}(G)$ is the Lie algebra of $G$.

I am really not an expert in Lie theory, so Thanks in advance for any help or suggestion.