Outpost was fine, but then said over capacity, and then prestige failed
So the outpost was apparently fine. I had set it to active and people had been raiding it.
But then I went in to collect some materials and also make a test run myself.
Then I tried to leave but I got a warning message (your base will be deactivated because it is over capacity and also some other problem).
So I thought I would do an early prestige, to see if that fixed it. I paid my 425, but the prestige got stuck in this four-square loading animation, and then I couldn't do anything. In the end, I had to kill the game and restart.
(At this point I was essentially in playtesting mode, rather than in building mode. As you can see, there is a Loadout screen on my screenshot.)
When I restarted the game, my synthite was taken, but the outpost was still on level 3!