Let $(X_t)_{0\le t\le T}$ be a stochastic process defined on $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous path. Assume that $X$ is square integrable in the sense that $$ \mathbb{E} \int_0^T X_r^2 dr < \infty $$ Can we expect that $$ \lim_{\varepsilon \to 0}\int_0^t \mathbb{E} \left[\frac{1}{\varepsilon}\int_r^{r+\varepsilon} (X_s-X_r)^2 ds \right] dr = 0 $$ for $0 < t \le T < \infty$?
I tried using DCT, so that $\int_0^t \mathbb{E}\left[ \lim_{\varepsilon \to 0}\frac{1}{\varepsilon}\int_r^{r+\varepsilon} (X_s-X_r)^2 ds \right] dr = 0$, but the dominated one is not available. Here is a similar question where we assume boundedness.
Any help would be appreciated.