Let $(X_t)_{t\in\mathbb{R}}$ be a (double-sided) Levy Process, i.e. $X_0 = 0$ almost surely, the functions $t\mapsto X_t\omega$, $\omega\in\Omega$, are right-continuous with left limits and the increments $X_{t_1+c}-X_{t_0+c},\dots,X_{t_n+c}-X_{t_{n-1}+c}$ are independent and their distribution is independent of $c$, where $n\in\mathbb{N}$, $t_0,\dots,t_n\in\mathbb{R}$ with $t_0<\cdots<t_n$ and $c\in\mathbb{R}$. Let $\tau$ be a $\mathcal{F}$-stopping time with $\tau\in\mathbb{R}$ almost surely, where $\mathcal{F}_t = \sigma(X_{s})_{s\leq t}$ is the natural Filtration of $X$.
Is the following statement TRUE or FALSE?:
For every $t\in\mathbb{R}$, $c>0$ and $B\in\mathcal{B}$, the Borel-$\sigma$-Algebra over $\mathbb{R}$, there is a measurable set $\Omega'$ with $\mathbb{P}\Omega'=1$, such that \begin{equation} \mathbb{P}\left[ X_{\tau+c}\in B \; \vert \; X_\tau \right]\omega = \mathbb{P}\left[ X_{t+c}\in B \; \vert \; X_t \right]\omega \end{equation} for every $\omega\in\{ \tau = t \}\cap \Omega'$.