I learned from Piotr Akhmetiev that $S^6$ contains two smoothly embedded $5$-spheres invariant under the antipodal involution that are not equivariantly PL isotopic, and the reference is [Lopez de Medrano's "Involutions on Manifolds"][1]. Of course they are boundaries of regular neighborhoods of a point (by the higher-dimensional Poincare conjecture) and hence also of tubular neighborhoods of a point (since there are no exotic $6$-balls). A more recent source that Akhmetiev mentioned is [Yu. Muranov's survey][2].




  [1]: http://www.zentralblatt-math.org/zmath/search/?an=0214.22501
  [2]: http://mi.mathnet.ru/eng/tm1075