Compact spaces, compact cardinals, and elementary submodels

If M is an elementary submodel and X a topological space, then XM denotes the set X∩M given the topology generated by the open subsets of X which are members of M. Call a compact space squashable iff for some M, XM is compact and XM≠X. The first supercompact cardinal is the least κ such that all compact X with |X|⩾κ are squashable. The first λ such that λ2 is squashable is larger than the first 1-extendible cardinal.