Pickands' constant $H_{\alpha}$ does not equal $1/\Gamma(1/\alpha)$, for small $\alpha$

Pickands' constants $H_{\alpha}$ appear in various classical limit results about tail probabilities of suprema of Gaussian processes. It is an often quoted conjecture that perhaps $H_{\alpha} = 1/\Gamma(1/\alpha)$ for all $0 < \alpha \leq 2$, but it is also frequently observed that this doesn't seem compatible with evidence coming from simulations. We prove the conjecture is false for small $\alpha$, and in fact that $H_{\alpha} \geq (1.1527)^{1/\alpha}/\Gamma(1/\alpha)$ for all sufficiently small $\alpha$. The proof is a refinement of the "conditioning and comparison" approach to lower bounds for upper tail probabilities, developed in a previous paper of the author. Some calculations of hitting probabilities for Brownian motion are also involved.