Fleeing demoralized unit move off the board?

[tsievers]

A demoralized unit fails a morale check in a hex near or adjacent to the board edge on which they entered. The only place it can flee to is off the board. 

Once off board is it considered in a safe hex? Can it continue to attempt recovery and if it succeeds re-enter the board?

Or, once the demoralized unit had fled off the board is it considered eliminated?

(Playing Pusan scenario #24 "Go Back", U.S. units ran into N. Korean units on one of the slope hexes only one hex away from the board edge. Several were demoralized and failed to recover, leaving them nowhere to flee but off the board edge they had entered just two turns before.)