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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". 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.