Compact spaces, elementary submodels, and the countable chain condition

Given a space 〈X,J〉 in an elementary submodel M of H(θ), define XM to be X∩M with the topology generated by {U∩M:U∈J∩M}. It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X=XM. Assuming (the Singular Cardinals Hypothesis) in addition, the result holds for any compact XM satisfying the countable chain condition.