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

In a recent paper by Gérard, Häfner, and Wrochna, 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 $\mathrm{\mid }a\mathrm{\mid }\ll M$. In this note, we extend this result to the full non extreme case $\mathrm{\mid }a\mathrm{\mid }<M$.