Frobenius liftable hypersurfaces

Let D be a reduced divisor in Pⁿₖ for an algebraically closed field k of positive characteristic p > 0. We prove that if (Pⁿₖ, D) is Frobenius liftable modulo p², then D is a toric divisor. As a corollary, we show that if there exists a finite surjective morphism f Y X onto a smooth projective complex variety X of Picard rank 1 such that (Y, f^-1 (D) ₑ₄₃) is a toric pair, then X is the projective space and D is a toric divisor.