A fiberwise analogue of the Borsuk–Ulam theorem for sphere bundles over a 2-cell complex

For a vector bundle α, let indα denote the largest integer m for which there exists a Z/2-map from Sm−1 to S(α). We prove that the equality indα=dimα holds for every vector bundle α over the complex Sn−1∪ken, where n⩾2 and k≠0, if and only if either k is even and n≠2,3,4,8 or k is odd.