Chernoff's density is log-concave

We show that the density of $Z = \argmax \{W(t) - t^2 \}$, sometimes known as Chernoff's density, is log-concave. We conjecture that Chernoff's density is strongly log-concave or "super-Gaussian", and provide evidence in support of the conjecture. We also show that the standard normal density can be written in the same structural form as Chernoff's density, make connections with L. Bondesson's class of hyperbolically completely monotone densities, and identify a large sub-class thereof having log-transforms to $\RR$ which are strongly log-concave.