Recently it was proved for 1 < p < ∞ that ${\omega }^m{(f,t)}_p$, a modulus of smoothness on the unit sphere, and $K̃ₘ{(f,t^m)}_p$, a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence ${\omega }^m{(f,t)}_p\approx K̃ₘ{(f,t^r)}_p$ does not hold either for p = ∞ or for p = 1.