This is something I've commonly heard said, usually by tire salesmen, but I do not buy. The difference in speed of the rotating tires between 6/32" is very small and likely less than many other factors such as alignment and tire inflation differences. The differentials are meant to handle small differences between the different tires rotation speeds and should be able to cope with this.
All this assumes that you are using the same size tires. Never mix tire sizes on a 4WD vehicle.
However, I'd recommend that you call Ford if they provide some sort of "answers line" and verify with them that there isn't anything unique in the design of the system that couldn't cope with this difference in tread depth on otherwise identical tires.
As an aside: We were once traveling through Montana and had a tire die. We limped into town and found a place that could replace the tire, but they were pushing for us to get all 4 matched tires, because we were driving an AWD car (Audi Quattro full-time AWD). I'm kind of particular about my tires and they didn't have what I would have wanted on it, so I just had them replace the one, so we could get home and deal with getting the right tire ordered in then. I was able to get them to replace just the one tire.
I later called the Audi Customer Service Hotline and asked about this. The person I spoke to said that as long as they were the same size tire, it was absolutely not a problem. She re-iterated that the tires must be the same size, but differences in the wear are fine. She obviously did not look this up or have to ask anyone, so it seems like a question they get a lot.