A paper just appeared on the arXiv that supports Jeff Lagarias's claim: "Upper bound on the packing density of regular tetrahedra and octahedra," by Simon Gravel, Veit Elser, and Yoav Kallus (medicine and physics researchers at Stanford and Cornell):
In this article, we obtain an explicit bound to the packing density of regular tetrahedra, namely $\phi < 1-\delta$ with $\delta = 2.6...\times 10^{-26}$10^{-25}$. [...] In order to obtain a bound to the packing density, we show the existence, in any tetrahedron packing, of a set of disjoint balls whose intersection with the packing is particularly simple, and whose density can be bounded below. The construction is such that the density of the packing within each of the balls can be bounded away from one. The combination of these two bounds gives the main result.
They modify their argument to obtain a $10^{-12}$ bound on regular octahedra.
