In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations and bundle projection between them. Under some mild topological assumption on the base space (of course verified in the case of manifolds) a fiber bundle always gives rise to a fibration; so in this context I consider fiber bundles as particular examples of fibrations.

My question: in which cases a general fibration turns out to be a fiber bundle?

EDIT: in this question on MO, they suggest that it is probably true that the projection of a smooth fibration is a submersion. Does anyone have a reference/counterexample for this?

If this happens to be true, then we can apply Ehresmann and obtain a fiber bundle (I am restricting to the case of compact smooth manifolds here). This would solve the problem at least in the case I was interested in.