Not sure if that's entirely true. I'd have to look it up.
And since I'm pretty lazy, I doubt I will. But I'd assume that, since the enchantment is still in play, that its static ability (i.e. the "constant" ability) would remain in effect. IIRC...
"Static abilities are 'on' when the card is in play and 'off' when it is not." (or something that)
However, the fact that you can only activate the activated ability before your combat phase prevents future abuse of it. If it weren't restricted to precombat, you could untap it, play the ability again, and potentially earn another combat phase (there are cards that do this. Godo, Bandit Warlord comes to mind), untapping it again. Makes the card seem less broken.