I'm getting a P0403 code (exhaust gas recirculation circuit malfunction) on a 2004 Mazda Tribute (3.0L V6). Upon the first instance of getting the code, I erased it to see if it would recur, as it wasn't causing any performance issues. It took a week or so for it to come back. At this point, I inspected the wires and contacts on the solenoid connector to see if there were any breaks or signs of corrosion, but at a glance they seem intact. I also checked the vacuum line, which was in pretty bad shape, so I replaced it. Following the initial inspection, I replaced the EGR solenoid and erased the trouble code. Took it for a test drive and the code didn't come back. I turned off the engine, cranked it back up, and the code came back about 5 seconds later. This is pertinent, as the code never recurs once erased while the engine is running, only at start up. Performed a vacuum test on the EGR valve while the engine was running and the engine stuttered, indicating that the valve is functional. I also tested it for vacuum with the engine off to be sure that the valve was capable of holding vacuum for at least a minute and it passed. I also tested the connector with a multimeter and a test light while the engine was running to see if the positive end was receiving 12 volts and the connector is receiving between 12-13 volts. Despite all the tests I've done getting positive results and installing a new solenoid, the code persists, so I'm kind of at a loss. I'm not sure how I should proceed save for taking the EGR valve and its associated lines off to check for carbon build up/blockages.  Any ideas on how I can narrow down the source of the problem further?

Thanks in advance!