If you either a) have decided to let fate decide for you if enough coolant is getting into cylinders to help you make the replace engine or head gasket or b) have already decided to run this engine at the risk of failure then replace the engine of it does, then put a cap on the radiator that opens at a lower psi so it will get out the TOP of the reservoir not the SIDE.

Then to the question: you have gas engine inferred and 3.0 I think I saw in a different post. I can check our cars if somebody else hasn't but photos better than description.

It seems like you have from and to backwards. I'm pretty sure the hot water goes to the heater core valves then the return would be from the firewall.

That said, some HVAC units have a rest mode that pumps heat from the engine core when the car isn't running some don't and I think that difference is what brings up the difference in the part exploded view.

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)