Indeed an interesting watch. Reference is 5681. They were most likely made in 3 batches of 100(?) pieces each with each batch slighly differing from each other (i.e. round pusher vs. square one). On the caseback the batch number and an individual number are stamped. In my case the batch is 23646. Movement is a modified 12.68Z. The chronometer function is interesting. In stead of the all hands at 0 untill you start all hands runs continously until you push the small pusher which will reset and restart the chrono. The crown has the usual functions of winding and setting the time. Case size is 37.5mm.

The 12.68 movement itself is not that rare but mostly seen in a smaller (32/34mm) case. As far as I can find about 10 examples have been on the market in the last decade. Phillips recently sold a similar example at their latest sale (https://www.phillips.com/detail/LONGINES/CH080016/43) and Hodinkee has one in their shop right now (albeit very pricy imho).