We did it in our Unimog a month or so ago.  We had no problems, and I don't think there would be anything a D4 with proper tyres could not handle.   The corrugations are bad, and go on for a long way, but the dunes are not that steep but a few are very soft at the crest.   Please don't be one of those idiots who just floors it up the dune digging big holes all the way up, just run low tyre pressures and take it nice and slowly.  Pump them back up after you pass that section if you want to go fast again, but corrugations are bad.  
 
  
We didn't start at Wiluna, but from Newman then on to the Talawana track getting on the CSR at Well 23.  You can get a fuel drop there, or if you came the way we did instead, then fill up at Parnngurr, and again at Kunawaritji, and Billiluna.  Fuel is expensive - $3.60 per litre for diesel.   We didn't need to fill up at Billiluna, but we were told there is a $50 call out fee as well if the guy is not at the pump when you arrive, not sure if this is true but you don't have a choice if you can't make the last 150km to Halls Creek.    
We took it pretty easy, about 100km per day, and we averaged 28lt/100km but then we are a 8 ton truck.    The longest stretch is Willuna to Well 23, about 700km, we didn't do that part, as we are too big for the first 150km anyway and we were just the CSR to get to warmer climates on our way home.  
Be prepared for a fair amount to scratches though, but we would have cleared it a bit for you.   
https://www.youtube.com/watch?v=7Z-3MYvFCcY
https://www.youtube.com/watch?v=Vg0YAYTRIzg
Full write up of our trip  here
Iain and Trish's Unimog Adventures
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