Is there a local martingale $X_t$ with $EX_t=0$ for all $t\ge 0$ but $X_t$ is not a true martingale?
I know two local martingales which are not martingales: $1/|B_t|$ in 3D, and you can also rescale a non-UI MG $\mathcal{E}(B_t)$ into $t\in [0,1]$, but neither of these satisfy the condition above. I saw necessary and sufficient conditions for a local MG to be a true MG in this question, and they don't seem to be implied by my condition above. But I can't think how to construct it. If my condition holds for bounded stopping times instead of only determinstic times, the conclusion follows from the converse to optional stopping theorem.
