I wouldn't be in a big hurry to steam clean. Material is glued to Luan with contact cement.
On my 2002 BF I washed the roof one time without having a vent completely shut by accident and shot water down the vent. It caused a lot of brown staining a couple of inches out from vent trim on inside ceiling. I'm pretty sure the brown stains bled thru from the wet Luan wood...
I left it for months trying to figure out what I should do, then wife said maybe try using Oxi-Clean laundry stain remover on it. I pulled the plastic trim off then used an Oxi-Clean spray bottle to wet material, then right away used a small sponge to work it so I wouldn't pull loops out of the fuzzy material like a brush might. I had my wet/dry shop vac there too and sucked the cleaner out of the material after working it. I went over it twice and later when it dried completely there was no trace of a stain or that the area of fuzzy stuff had been worked on...only thing I would be careful about is if the whole ceiling is too dirty then the cleaned area might show as a whiter patch.
