Let $X_t$ be univariate stochastic process. Assume $X_t$ is conditionally stationary that is for any $n\in \mathbb{N}$ and time indices $t_1,\ldots, t_{n+1}$, $t_{i}<t_{i+1}$ and shift $\tau$
$$F(X_{t_{n+1}}|X_{t_{1}}\ldots,X_{t_{n}})=F(X_{t_{n+1+\tau}}|X_{t_{1}+\tau}\ldots,X_{t_{n}+\tau}),$$
where $F$ is the distribution function. Assume the conditional mean function exists so that
$$\mathbb{E}(X_{t_{n+1}}|X_{t_{1}}\ldots,X_{t_{n}})=\mathbb{E}(X_{t_{n+1+\tau}}|X_{t_{1}+\tau}\ldots,X_{t_{n}+\tau}),$$
Does this imply that $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exists? If not can we add some further assumption to guarantee the limit exists?