The wiring for the N239 solenoid is real simple. If you have a Bentley manual, turn to page 97-692, section number 66/8.
The purple/grey wire goes directly back to terminal 81 on the ECU. The ECU grounds this wire to activate the N239 solenoid. That could be checked for continuity.
The yellow/black wire is the 12v supply. It is connected inside the wiring harness to the yellow/black wires on the N75, N18 and MAF sensor. Those all go back to Fuse 34.
So, I assume you should see battery voltage on the yellow/black wire when the ignition is on. BTW, the yellow/black in Terminal 1 on the connector, same as on the other 3 components listed above.
If you're not getting 12v at the N239, the issue could be farther inside the harness where all those yellow/black come together.
Your 3v seems low, I assume that should be 12v. When the ignition is turned off you should see a ground signal on the purple wire (to activate the solenoid), with the purple staying at 12v.