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If it was only doing it when connected I'd suspect either an ECM or wiring fault but as it's all the time it sounds like you've got a set of faulty/seized injectors as they *should* be closed until the ECM triggers them.
Send them away for testing/cleaning as they *may* be salvageable
Try putting a multimeter across the two terminals on each unit and make a note of each resistance value - If they're in a good electrical condition/state they'll be around 13.5-15.5 Ohms each. If they're all showing higher then i'm afraid they're fubar'd but as above, possibly refurbish-able (If that's a real word)