PoT drop rate

I'll add another question to the thread:

Me and my friend were arguing today, he insisted that PoT drop chance doesn't increase as you go deeper into ToT. I believe it does, in fact, increase.
Which is right?

I don't have any hard evidence, but while doing the ToT quests, I got 4 PoT's in the memory lanes and none in the qualm lanes. That would seem to suggest otherwise, but I could have just gotten lucky and then unlucky.