I have a question about Feller property of reflecting Brownian motions.

Let $D \subset \mathbb{R}^2$ be a domain. Assume $D$ is represented as \begin{equation*} D=\{(x,y) \in \mathbb{R} \times \mathbb{R} \mid |y|<H(|x|)\}. \end{equation*} Here, $H$ denotes a smooth function on $[0,\infty)$.

Let $X_t$ be the reflecting Brownian motion on $\overline{D}$ and $X_t^1$ be the first coordinate of $X_t$. According to Pinsky's heuristic argument (see p.4 of 1), the behavior of $\rho(t)=|X_t^1|$ over the long run should be like the behavior of the one dimensional diffusion generated by \begin{equation*} \frac{1}{2}\frac{d^2}{d \rho^2}+\frac{C H'(\rho)}{H(\rho)(H'(\rho)^2+1)}\frac{d}{d \rho}, \end{equation*} where $C$ is some positive constant. Hence, if $H'(\rho) \to 0$ as $\rho \to \infty$, the above diffusion is not much different from the one generated by \begin{equation*} \frac{1}{2}\frac{d^2}{d \rho^2}+\frac{C H'(\rho)}{H(\rho)}\frac{d}{d \rho}. \end{equation*}

**My question**

Let $H=\exp(-x^4)$. If we believe the above heuristic argument, the inner drift of $|X_t^1|$ becomes stronger as $X_t$ goes to infinity. Hence, I think $X_t$ is not Feller process by means of $p_{t}(C_{\infty}(\overline{D})) \subset C_{\infty}(\overline{D})$. But I couldn't prove this claim which is seemingly correct.

Do you know how to prove this?