Suppose I have two inequivalent fibered knots in a homology sphere. When I say 'inequivalent', I mean that there is no orientation-preserving homeomorphism of the space that takes one to the other. Can they have orientation-preservingly homeomorphic complements?
I guess in general the answer is 'we don't know, but we think not', which is the oriented knot complement conjecture. What I'm asking here is whether fiberedness makes things much easier. It intuitively feels to me, that fibered knot complements are quite 'rigid', so perhaps it should somehow be easier to resolve the conjecture in this case.