I've been doing this for just over 2 weeks, and so far this experiment has been a success. Here is my sleep chart for the past 9 days:

I copied this in the morning so the afternoon sleep from today is missing (it hasn't happened yet).

As you can see, there have been quite a few days where I've woken up late for one reason or another. The latest I woke up was on Thursday when I woke up at 7:30am. Each time this happens I think "I won't be napping today" but so far I haven't missed a nap.

I'm still a little worried that over time my sleeping pattern will start to shift, so I'm going to keep an eye on things until I've been doing this for at least a month. I'm not sure if I'll do another update on this, but you can always check on my sleep chart because it's pretty obvious if I go back to using the alarm as the wake up time would be the same every day.