One of the logistical issues is refueling the plane twice a day. As previous record-holders did, the Pipkins will have to fly close enough to the ground to allow the fuel to be pumped from a tank in the back of a truck moving at a matched speed.
Wouldn't it be safer (and easier) to install an air-to-air refuelling facility on the aircraft? Admittedly difficult in the administrative sense, but flying low and slow doesn't sound like a good idea.
Also I'm curious about how they're going to stock up on food, and how they plan to relieve themselves when nature calls. If they were flying a twin, the logistics wouldn't seem so impossible, because they'd have a little bit more room to play with for cargo, on-board facilities, etc. But in a Cessna 172? Sounds like its going to be an uncomfortable flight.
That said, I'm full of support for the venture; and wish the two the very best of luck. It'll be really interesting to read how they get on.