The fpr controls fuel pressure by an internal spring acting against a diaphragm. The fuel pressure in the rail is varied by applying vacuum or boost to the diaphragm in the fpr. When the engine is not even running, the fpr sees atmospheric pressure.
If the fpr simply opened because the engine was not running, it is unlikely that the engine would start because there would not be enough fuel pressure in the rail to squirt properly through the injectors.
According to my workshop manual, cold start problems can be caused by:
- cold start enrichment from the engine temp sensor is not working
-- try starting with the engine temp sensor disconnected
-- check engine temp sensor resistance
-- check engine temp sensor output voltage
- idle speed controller is not working
- start time enrichment is not working
- cranking speed too slow
Then combine with the poor cold idle:
- idle control system not working correctly
- cold start valve not working properly
- engine temp sensor not working correctly.