I don't believe there would be near enough room to fold and stow a queen size 4" memory foam mattress in our sofa. The air mattress folds and stows with 4 straps, and must be folded correctly and strapped properly so as not to interfere with folding up the sofa. There's a bit of storage under the sofa, but not near enough for this IMHO.
Pat & Peg (and our golden retrievers)
2016 London Aire 4519