I see something similar on my setup here with 2.1. Well, I used to see it: every time I connect the screens on Mavericks I get a different behaviour.
The screen that is blank was actually appearing on the desktop: it was possible to manipulate it in the Arrangement tab of the Displays system preference.
Is it happening the same in your case?
If it is, can you also please report (using 2.1 final) the name of the blank screen as it is shown in the title of the Displays preference for it? (Use Gather Windows to see it)
The logs collected through
http://www.displaylink.com/support/ticket.php?id=269 would be helpful as well.
To give you some context, we have some workarounds for the Mavericks multiple display behaviour in 2.1 that are not in the beta. For most people they seem to be beneficial but for some definitely not! I hope we'll be able to find more comprehensive workarounds while we wait for an OS fix.
Regards,
Carlo