Integral of product of Hermite functions with rescaled weights.

Question

Let $$h_{k}(x)=c_{k}(-1)^k e^{\frac{x^2}{r^2}}\frac{d^k}{dx^k}e^{-\frac{x^2}{r^2}}$$ be the standard Hermite polynomials, rescaled with a given parameter $r>0$. The normalizing constant $c_{k}=(\sqrt{\pi}r2^k k!)^{-1/2}$ ensures that $h_{k}^2e^{-\frac{x^2}{r^2}}$ integrates to one. I'm trying to understand the asymptotic behaviour of $$C_k = \int_{-\infty}^\infty |h_k(x)|^4 e^{-\frac{x^2}{b^2}}dx$$ as a function of $k$. By looking around at some results (in particular http://www.math.u-psud.fr/~wang/IntHerm.pdf), it seems that $$C_k \leq \delta^{2k}$$ for some constant $\delta>0$ when $b=r/\sqrt{2}$. Can we say the same when $b$ is higher for example?