there was an update citing the issues described in this very thread:

・ Error 4004, 5003, and 5006
These errors occur when your connection to the login management server times out as a result of waiting in a login queue for extremely long periods of time.

so basically, if the Q is too long, you're fucked, coz by the time it will be your turn, the login server will time you out, coz you know, SE don't have enough money to increase HW capacity.. Thank you, TYSM for informing us that we basically stand no chance to actually log into the game if the queue is too large, I feel very enlightened right now!!! Maybe find a solution instead of saying what we all know.