  We prove that if two topologically free and entropy regular actions of
countable sofic groups on compact metrizable spaces are continuously orbit
equivalent, and each group either (i) contains a w-normal amenable subgroup
which is neither locally finite nor virtually cyclic, or (ii) is a
non-locally-finite product of two infinite groups, then the actions have the
same sofic topological entropy. This fact is then used to show that if two free
uniquely ergodic and entropy regular probability-measure-preserving actions of
such groups are boundedly orbit equivalent then the actions have the same sofic
measure entropy. Our arguments are based on a relativization of property SC to
sofic approximations and yield more general entropy inequalities.
