On fares matters at least, they can't possibly. We call it "collusion".
True, there are circumstances in which they are not allowed to work together! Ironically, there is evidence that sometimes they do, when they are not allowed to (e.g. the Newark to London routeing guide change scandal).
In contrast, there are some circumstances when they
should be working together (e.g. helping passengers when a delay has occurred) but sometimes do not.
They really should not be making such bold claims if it doesn't reflect reality!