Our resourceful beta testers have managed to get YNAB working under Linux by using WINE. We don't officially support Linux because it proved to be quite difficult and time consuming for our support staff to debug issues on the various flavors. In addition, we run on AIR, which is already incompatible with Linux in many ways, and might break further in the future. However, we are thrilled that it looks like our existing Linux users - the adventurous ones at least - might have a way to upgrade to YNAB 4.
I'm going to make this thread an announce post and invite WolverineFan, the fellow that got this working, to post his instructions working below. (He even wrote a script to automate it). Enjoy!