Let X be a simply connected finite CW complex, \xi and \eta  vector bundles over X of the same dimensions and their dimension is big enough, so they are stable bundles. Let p be a prime.
Are the following two conditions equivalent?

1. J(\xi) and J(\eta) as elements in J(X) have equal p-primary components.
(That is there is a q not divisible by p such that q(J(\xi) - J(\eta) ) = 0.)

2. The sphere bundles S(\xi) and S(\eta) are fiberwise p-equivalent.
(That is there is a fiberwise map S(\xi) \to S(\eta) that induces isomorphism in homologies with Z/pZ coefficients.)