I've been doing some more research on the internet and I'm starting to
think that the problem is not aphids at all.

The plants (leaves and stems) have brown and black spots (not more than
1mm diameter each) and they don't move. They are stuck underneath the
leaves and along the stems. They can also be found on top of the
leaves. The spots are not actually within the leave membrane but on the
outer side... sort of miniscule domes.

As I said, I've pruned a plant by removing all leaves and treating the
stems with a soap mixture and also with pesticide. When new leaves
grow, the spots come back again.

Could it be a different kind of insect or a fungus infection?

Thanks,
Max