There have been several recent advances on packing regular tetrahedra in $mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there exists an arrangement of tetraheda filling about 72% of space. This has been improved in a series of papers, and the latest result of which I am aware is Elizabeth Chen's record of 85.63%. (A NYTimes article summarizing the history of the problem can be found here.)
My question is does anyone know of any upper bounds, either published or unpublished? I saw a colloquium by Jeff Lagarias, and he said someone was claiming that they had proved something like $1 - 10^{-26}$, but that it was still unpublished.
(A compactness argument gives that since regular tetrahedra don't tile space the maximum volume is strictly less than one, but this argument does not give a quantitative bound.)
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