Last night I finished building enough stability in the final 3 timelines so that all I had to do was complete the 3 story echos in each to unlock Empowered monoliths. I did 1/3 story echos on one timeline and realized it was late and I was tired so I called it a night.
Today I logged in on my lunch to do the remaining 2/3 on that timeline and finish that timeline. However, when I logged in the timeline showed no progress on the 3 story echos. I thought maybe I logged out too quickly after completing the echo and something didn’t have a chance to save on the server. I completed 3/3 story echos and finished the timeline. Then since I had some time left on my lunch I decided to experiment. I went to the next timeline and did exactly 1 story echo. To give the game time to ensure it communicated with the server, I went to the observatory to purchase some Prophecies. I bought a few and logged off.
When I logged in after work my progress on the story echos was once again reset, however, the Prophecies I purchased were still there. This time I decided to do 1 story echo and then one regular reward-based echo and log off. Then I immediately re-opened the game and the progress on the story echo was still there.
So as far as I can tell, if the last echo you do is a story echo and it’s not the final 3/3 echo for the timeline you will lose the progress on that timeline’s stories.