Lipschitz-on-bounded-sets characterization for C_f on bv_p(ℝ) with p>1

Determine whether the composition operator C_f: bv_p(ℝ)→bv_p(ℝ) is Lipschitz continuous on bounded sets if and only if the derivative f′ is Lipschitz continuous on bounded sets when p>1.

Background

For BV_p[a,b] and for bv_1(ℝ), a classical equivalence holds: C_f is Lipschitz on bounded sets if and only if f′ is Lipschitz on bounded sets. Extending this equivalence to bv_p(ℝ) with p>1 remains unresolved, again reflecting the complications introduced by unbounded sequences in bv_p when p>1.