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)red) is a toric pair, then X is the projective space and D is a toric divisor.