Achronal averaged null energy condition for extremal horizons and (A)dS

We prove the achronal averaged null energy condition for general quantum field theories in the near horizon geometry of spherical extremal black holes (i.e. ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times S^{d-2}$), de Sitter and anti-de Sitter. The derivation follows from monotonicity of relative entropy after computing the modular hamiltonian of a null deformed region. For incomplete (but maximally extended) achronal null geodesics in ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times S^{d-2}$, we prove the positivity of a different light-ray operator for arbitrary CFTs. This agrees with a constraint recently derived for the Lorentzian cylinder.