That's what I thought.

I am exiting the EU though, so I will be going through passport control. What I saw last time is that I appear to need to exit Schengen, re-enter at K, L, and repeat the process on the way back as my flight is in H.

unless there is a transfer passenger path I am unaware of.

The info booth at the airport was insistent that it is possible. So I have a few hours to kill, so why not try?