For over 15 years, I had sand toys and played in the sand. Airing down was the norm. On our trucks, driving on the beach we had the widest tires we could buy. Seldom did we air down. Super soft sand maybe air down. When we played with our toys, we aired them down.
What I find interesting is that people say Dually's are no good in the sand. Yet they run singles and air down for a bigger footprint and flotation. Looks to me the dually has twice the flotation as the single. Wide tires with paddles, floatation and traction would be ideal, but the paddles would not do well on the road.
I have ran the Rubicon Trail in my Samurai with the tires aired up and aired down. I honestly cannot say I experience major difference. I do think aired down a little reduces the likelihood of tire punctures.
Keep in mind I am citing my experiences, not someone else's. I will not get into a long debate on the subject, especially on this thread.