The tilting property for F_*ᵉ OX on Fano surfaces and threefolds

Let X be a smooth variety over a field of characteristic p. It is a natural question whether the Frobenius pushforwards F_*ᵉ OX of the structure sheaf are tilting bundles. We show if X is a smooth del Pezzo surface of degree ≤ 3 or a Fano threefold with vol(KX)<24 over a field of characteristic p, then Extⁱ(F_*ᵉ OX,Fᵉ_* OX)≠ 0 and thus F_*ᵉ OX is not tilting.