Correct me if I'm wrong, but isn't it disputed, with heavy border tensions and all? I'm not claiming India doesn't control it, and as a Swede I don't know much. But that's about the one thing I thought I knew about Kashmir.

Yes, heavily disputed, to the point that there are widespread campaigns on the internet to normalize the various sovereignty claims under the guise of day-to-day conversation about those locales (e.g. conspicuously referring to them as being "in India" or "in Pakistan" or "in China"). For example, see the Google maps reviews for any landmarks in the area.