John Gustafson is on the board of Rex Computing, a small semi startup. They claim to have taped out last year, but I don't know if the chip has been validated or if it had posits in silicon, but I know that is one of their longer term goals.
I find Gustafson's UNUMS 2.0 more compelling than the first version.
Founder of REX Computing here, we taped out in July of 2016 and got our silicon brought up and working back in February. I gave a talk at Stanford showing our hardware and a tiny bit of software: https://www.youtube.com/watch?v=ki6jVXZM2XU
Type 2 unums are pretty much entirely deprecated by type 3 unums (now given the name 'posits' as referred to in the OP's linked paper)... they are basically superior in every way, and I recommend watching John's February 2nd talk at Stanford.
I find Gustafson's UNUMS 2.0 more compelling than the first version.