When trying to auto-update from 3.7 to 3.7.3, I go through the whole process, click “Install and Relaunch”, then the app quits and doesn’t reopen it. When I reopen it manually, it’s still at 3.7.
Seems like I’ve had this issue with TaskPaper before, but it’s been a while and I can’t remember.
Here’s Console.app output for one full cycle of launch, attempt to update, and quit (filtered by the search string “TaskPaper”). There doesn’t seem to be anything relevant in