I've completed this quest several dozen times and lately the trend seems to be to have people not move when they log in as it starts the timer. I'm not sure where this comes from but I've completed the quest with someone moving or not moving as the case may be. For that matter I've completed it with a dog breaking its charm within 20 seconds and with no dogs breaking until the hound is dead. The only thing that seems to matter is to keep healing the puppies no matter what.
This got me wondering. Doesn't a quest start as soon as you enter? Is the hound run timer a DDO urban legend?