I have deducted that three things are possibly causing this failure on the 10th turn (when it should end):

New Event: An Unexpected Guest 

(+2 Kalgeros on a [Specific] system)

or

The Good Life Quest Completion (Bring the Lost Horatio Ship into a system populated by the player)

or

Loading into the save game while the Those Darn Kids quest is pending (like mentioned on another thread).

In hte process of loading multiple times to try every possibility, this latest time made the quest succeed. The only thing worth mentioning is that, instead of clikcing the button to move all the fleets, I just clicked end turn, which caused both of the end of quest triggers to occur at about the same time.