Why would a leaky check valve cause these symptoms? When the ignition is turned on the pump runs for a few seconds and that should re-pressurize the system. FWIW I had the same problem on a '84 volvo, but that car had K-jet injection. I can't remember now, but I don't think I was ever able to resolve the issue, but it was exactly the same, I had to keep it floored and then it would run on just a few cylinders for a while, then eventually clearing up.

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Csaba
1972 280SEL 4.5, silver