(For completeness I should mention that pypy3.5 works, but pypy3 doesn’t. I assume pypy3.5-5.10.0 would also work, but I haven’t needed to reach for it.)
Experiments show that I can request pypy2.7-6.0 and pypy3.5-6.0.
I still cannot request pypy2.7 to get the latest available version, and while I can request pypy3.5, that doesn’t get me 6.0, it gets me 5.10.1.
(I’m using http head https://s3.amazonaws.com/travis-python-archives/binaries/ubuntu/16.04/x86_64/pypy${version}.tar.bz2 to see what works and what doesn’t.)