Two years after starting this thread I have finally solved it.
The problem was that there are a few more AE-related tables in the PCM, which are not present in the HPtuners software.
This is what I see in a different tuning application:
Untitled.png
The green maps are the ones that we get in the VCM Editor, and which I had edited but didn't seem to work. There are ten maps, plus one table that defines the relationship between modes and maps (i.e. which map should be used in which mode). We know them. They're these ones:
Untitled1.png
Then we have the red maps, which are missing from HPtuners. These seem to define a desired position of the valves for each mode under certain circumstances. In my case the desired position in neutral and the desired position at idle were set too low. Even though in the VCM Editor I had set the valves to be 100% open in Normal mode at any rpm, they were only 60% open at idle because that's how the "idle" table (which I didn't see in the VCM Editor and didn't even know existed) happened to be set from the factory.
Now everything is working as it should. I'm finally happy.
