The generalized stable equivalence problem

In this note we study the following problem. Let k be an algebraically closed field and X be an affine variety over k. Suppose that H1,H2⊂X are two hypersurfaces such that there exists an automorphism f of X×kn satisfying f(H1×kn)=H2×kn for some n>0. Does this imply that there exists an automorphism of X such that ? We give an affirmative solution if one hypersurface is not k-uniruled and a counterexample in general.