I'm going to try that next. But if the derailleur should just "pop out" due to the spring, isn't it odd that it only seems to delay in shifting from the 5th to the 6th cog and not really the others? Well, from the 6th to 7th is a bit slow, too, but then after that it seems to be mostly fine.

But I'll clean and lube the derailleur first to see if that does the trick.