Hadamard property of the Unruh state for massless fermions on Kerr spacetime: the large a case

In Gérard et al. (Ann Sci Ecole Norm Sup 56:127–196, 2023), the Unruh state for massless fermions on a Kerr spacetime was constructed and the authors showed its Hadmard property in the case of very slowly rotating black holes \({\left| a \right| }\ll M\) $\left|a\right|\ll M$ . In this note, we extend this result to the full non extreme case \({\left| a \right| }<M\) $\left|a\right|<M$ .