Assume a filtered probability space, where the filtration is the augmented filtration generated by a Brownian motion.
More precisely, define the set $$ \mathcal{N} = \{ A \subset \Omega \mid \exists B \in \mathcal{F} : P(B) = 0, A \subset B \} $$ Then define $$ \mathcal{F}_t = \sigma( \mathcal{N}, W_s , 0 \leq s \leq t ), \quad t \in [0,T]. $$
Let $M = (M_t)_{t \in [0,T]}$ be a square integrable martingale. Then, we can choose a cadlag modification $\tilde{M}$ of $M.$ By the martingale representation theorem $\tilde{M}$ is continuous up to indistinguishability.
Does it follow that $M$ is continuous up to indistinguishability?