Essentially, that connector is for various wires from the chassis to the engine block. Wires directly from the ECU to the block (like the wires that run the plug coil packs, fuel injectors or various sensors) do not pass thru that connector. That connector is more for switched power feeds to various appliances on the block, like the air pump, A/C clutch or the coil packs themselves.

Here are the pinouts, I am sure of all but pin 2:

Pin 1* - Air pump relay control line - operated by ECM via Fuse 2 on Relay Module.
Pin 2 - SRT ONLY - either supercharger clutch (Probably) or Intercooler pump (probably not, since it's not on the engine block).
Pin 3 - A/C compressor clutch - fused via fuse 35 or 37 (manual does not tell us).
Pin 4* - Feeds coil packs and noise suppressors via Fuse 11.
Pin 5* - "Battery/Alternator" indicator - goes from alternator tertiary output to the instrument panel.
Pin 6 - Engine starter motor solenoid - fed from pulse module, not directly fused.

Those with * are those I'd suspect in your case as these are 'battery hot' feeds - other pins are switch off with the key off, you can verify this with a voltmeter. And you SHOULD verify this, as there are errors in the manual.

A leaky coil pack pass transistor would certainly cause this, pulling fuse 11 would tell you if that was the case.