I think I would check the fuel pressure at the injectors right after the engine has stalled after its 10 minutes of running. It sounds like it is starving for fuel. If the pressure is low, there is an obstruction somewhere between the pump and injectors. If this is the case, the fuel pump will probably be noisy, also. Another possibility is that the fuel distributor is gummed up from sitting, and the internal "valves", which control fuel flow to the injectors as the airflow meter plate moves, are sticking, clogged, etc.
A third possibility is water in the fuel, and a final possibility is some kind of ignition problem. I would check the timing with the engine running, and the strength of the spark after the engine has done its dying routine. Hope this helps lead you to the problem, as it sounds like a neat car.