The new tubes and rim tape arrived today. The extra wide rim tape covers more than the entire rim and it took a while to install without any folds. The way the tubes are installed now there is no touching with anything that isn't tire or rim tape and the rim tape is entirely within the tire.
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I also inflated the tires to a much higher pressure than before. I'm about to go on a 60km ride from Amsterdam to the coast and back, so hopefully they hold up!