In Hatcher's Algebraic Topology book it is noted after 4.61 that:
fiber preserving map + homotopy equivalence $\Rightarrow$ fiber homotopy equivalence.
Could there be two fibrations over the same base space where the total spaces are homotopy equivalent, but there is no fiber homotopy equivalence between them? (and therefore also no fiber preserving map)
If so, I would be glad to have a simple example.