Exponentially small splitting of separatrices near a period-doubling bifurcation in area-preserving maps

We consider the conservative H\'enon family at the period-doubling bifurcation of its fixed point and demonstrate that the separatrices of the fixed saddle point nearing the bifurcation split exponentially: given that $\lambda_+$ is the smaller of the eigenvalues of the saddle point, the angle between the separatrices along the homoclinic orbit satisfies $$\sin \alpha = O(e^{-{\pi^2 \over \log |\lambda_+|}})+ O\left( e^{-2 (1-\kappa) {\pi^2 \over \log |\lambda_+|}} \right),$$ for any positive $\kappa<1$.