Propriety of the reference posterior distribution in Gaussian Process modeling

In a seminal article, Berger, De Oliveira and Sans\'o (2001) compare several objective prior distributions for the parameters of Gaussian Process regression models with isotropic correlation kernel. The reference prior distribution stands out among them insofar as it always leads to a proper posterior. They prove this result for rough correlation kernels - Spherical, Exponential with power $q<2$, Mat\'ern with smoothness $\nu<1$. This paper provides a proof for smooth correlation kernels - Exponential with power $q=2$, Mat\'ern with smoothness $\nu \geqslant 1$, Rational Quadratic - along with tail rates of the reference prior for these kernels.