Noninteger sandwiched Rényi quantum null energy condition

Prove the Rényi quantum null energy condition for noninteger b1>1 in general, namely establish nonnegativity of the diagonal second null shape variation of the sandwiched Rényi divergence for all admissible states and relevant quantum field theories.

Background

The paper distinguishes the sandwiched Rényi divergence, corresponding to the slice z=b1, from the more general b1z family. Although the diagonal Rényi QNEC is known in certain free-field and algebraic settings, the authors state that its validity for noninteger b1>1 remains unresolved in general.

The unresolved issue concerns positivity of the diagonal second null shape variation, (D_b1)''_{++}[V;y]e50, for the sandwiched Rényi divergence beyond the cases already established in the cited literature. The paper derives a quasi-local formulation conditional on this positivity but does not prove it generally.