There is a theorem as follows:

**Theorem.** *Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \mathbb{E}M_t$ is right-continuous (which is always true of martingales on right-continuous filtrations). Then there is a RCLL (Cadlag) modification of $M_t$.*

Question.If I change "right-continuous" and "RCLL" to "continuous", is this still true? In other words, if the filtration is continuous and the map $t\mapsto \mathbb{E}M_t$ is continuous, can I get the stronger conclusion that there is a continuous (not just RCLL) modification?If it is true, is there a reference (or obvious proof)? If it is false: Is there a nice counterexample? Are there known conditions on the filtration that would guarantee the continuous modification?

I think I have a proof for martingales (it involves algorithmic randomness, so it is not at all standard), but since I cannot find this written anywhere, I am worried I might be mistaken. Also, I know it is true for martingales on the augmented filtration of Brownian motion, but that proof goes through the Martingale Representation Theorem (I believe) and seems like that is overkill (again making me worried I am missing something).

**Notes:** This question started out as a question on math.stackexchange. After a few weeks with no answer, I moved it here. Also, my question looks similar to another question on Mathoverflow, but they are different.