Let $G$ be a complex reductive algebraic group acting on a complex variety $X$ (not necessarily projective) with $\text{Pic}^G(X)$ finite dimensional (for simplicity). For a fixed $x \in X$ define $$P^s(x)=\{L \in \text{Pic}^G(X)_{\mathbb{Q}} : x \in X^s(L)\}$$ and $$P^{ss}(x)=\{L \in \text{Pic}^G(X)_{\mathbb{Q}} : x \in X^{ss}(L)\}.$$
Question: Is $P^s(x)$ open in $\text{Pic}^G(X)_{\mathbb{Q}}$ and is $P^{ss}(x)$ closed?
For $X$ projective and when restricting to the cone of ample $G$-linearized divisors, this should be true by the Hilbert-Mumford numerical criterion. I would be happy for either a counter-example or for an affirmative answer (possibly with stronger conditions on $X$).
