The picture above is not the EGR valve .

It is the EVAP solenoid. It too can cause rough idle, but as the OP says his trouble starts after three minutes that leads me to believe the car changing from closed loop to open loop is involved in the problem. The EVAP solenoid will open/close per the PCM after the car is up to operating temperature, has operated for awhile and then goes to idle. The EGR valve should always be closed at idle.

However, plugging the line going to the throttle from the solenoid as you indicate will differentiate between either problem.