Consider a diffusion process $(X_t)_{t\geq 0}$ satisfying mean-square ergodicity \begin{align*} \bar X_t := \frac{1}{t}\int_0^tX_t\,dt \;{\stackrel{L^2}{\longrightarrow}}\;\bar X,\quad t\to\infty, \end{align*} As an example, we can think about an Ornstein-Uhlenbeck process \begin{align*} dX_t = a(b-X_t)\,dt + \sigma\, dW_t, \end{align*} for which it is easy to show that $\mathbb{E}[(\bar X_t-b)^2]\to 0$, as $t\to\infty$.
Do this imply some kind of a uniform tail convergence for the time average? For example, \begin{align*} \mathbb{E}[\sup_{t\geq t_0}(\bar X_t-\bar X)^2]\longrightarrow 0,\quad t_0\to\infty, \end{align*} or that for every $\epsilon>0$, \begin{align*} \mathbb{P}[\sup_{t\geq t_0}(\bar X_t-\bar X)^2\geq \epsilon]\longrightarrow 0,\quad t_0\to\infty, \end{align*} or something weaker than that?