Let $X$ and $Y$ be two random variables and let $ \leq_{\mathcal{F}} $ be an integral stochastic order as defined in the literature of Müller et al. Let $ \mathcal{R}_{\mathcal{F}} $ be the corresponding maximal generator of that order. I wonder whether \begin{equation} \mathcal{R}_{\mathcal{F}} \cap C^{\infty} \subset \mathcal{R}_{\mathcal{F}\cap C^{\infty}} \end{equation} holds or whether there is another relation between the two sets.
Relevant definitions:
1) $ X \leq_{\mathcal{F}} Y \quad :\Leftrightarrow \quad \mathbb{E}f(X) \leq \mathbb{E}f(Y) \quad \forall f\in \mathcal{F} \quad $ for which the expectations are finite
2) $ \mathcal{R}_{\mathcal{F}} := \{ f \,| \, X\leq_{\mathcal{F}} Y \implies \mathbb{E}f(X) \leq \mathbb{E}f(Y) \} $
Thanks!