I logged in, did an echo, and left it.
I opened my inventory and noticed that my relic was missing from my inventory.
I emptied my inventory (clicking “Transfer Crafting Items” and then shift-clicking items into appropriate tabs in my stash).
Then I opened the next echo, immediately saw a shrine that dropped affix shards, and clicked on it. I tried to pick up the affix, but I got the error “No Space In Inventory” (or something very similar). I confirmed that my inventory was empty, and immediately logged out of the game.
I then logged back in, and attached to my cursor was the relic that was missing from my inventory. The echo was not listed as failed. I completed the echo and was able to loot items during it.