I was the Company Manager for a theatre on a COST contract and am now an AEA SM there; we provide housing at no cost to the actor.
However, I looked in my rulebook, and according to rule 29(A)(3) (and I'm paraphrasing; see page 33 of the rulebook), for a non-touring production the producer may either pay per diem as specified in rule 49 and have the actor pay for housing in accordance with rule 29(A)(2) or provide housing. I'd be interested to know what you find out from your email to Equity.
If you have any other questions about the COST contract, please feel free to PM me. I've become pretty familiar with it from a producer's and a stage manager's point of view.