There are some ideas floating around, but none have been confirmed. We're starting to lean towards an invertebrate, rather than a vertebrate. And our strongest leaning is towards a fossil barnacle (at least as of 9 p.m. on 29 March 2017). [Note: Although some aspects of this object look barnacle-like, others do not, so we're still looking for assistance with the identification.]
We're continuing to work on the identity of this object, so I thought I'd share a few more images, in case they're helpful.
A nice scan of two different sides:
I need to take a better picture of the wider, open end, but here's one view of the interior edge (looking down into the hollowed portion at the top of the first photo):
The photo above was taken under a microscope. I was so intrigued by the patterning that I zoomed in even closer:
I'm not sure whether this helps with the identification — it might just be a pattern that developed while the object was being fossilized.
I'll provide updates if we learn more...