As an alternative solution, I changed the owner of ~/.conda to myself:
sudo chown -R $USER ~/.conda
I think the root cause of the problem (at least in my case) was that ~/.conda was owned by root for some reason. Other solutions suggest making its access more permissive (open writing to the group or even to all) as a workaround. While technically that should work, I think changing the owner (and keeping existing permissions) is a safer approach.