If the transfer failed, you will see an error or prompt icon next to the transfer. On success the transfer turns gray and goes away eventually.
I believe this is the indication you ask for.

This columns should be empty on waiting and during transfer.
After a completed transfer it should show the exact time the transfer finished successfully.
If the transfer failed it can show either "Failed" or be empty.

I need this to know whether the document I am editing has been saved on the server and whether it was successful and when it was finished.
So I know when I can use/execute (if it is a script file).