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?
$J(\xi)$ 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 $q(J(\xi) - J(\eta) ) = 0.$)
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.)