correlation between two functions of a random variable

Question

Suppose random variable $X$ has distribution depending on $\theta \in S$, and for some function $g(X)$ is uncorrelated with any function of $X$ at some $\theta_0 \in S$, i.e., $E_{\theta_0} (g(X)h(X)) = E_{\theta_0}(g(X))E_{\theta_0}(h(X))$, for any function $h$. Can we conclude that $E_{\theta} (g(X)h(X)) = E_\theta (g(X)) E_\theta(h(X))$ holds for all $\theta \in S$ ?