  We say that two free probability-measure-preserving actions of countable
groups are Shannon orbit equivalent if there is an orbit equivalence between
them whose associated cocycle partitions have finite Shannon entropy. We show
that if the acting groups are sofic and each has a w-normal amenable subgroup
which is neither locally finite nor virtually cyclic then Shannon orbit
equivalence implies that the actions have the same maximum sofic entropy. This
extends a result of Austin beyond the finitely generated amenable setting and
has the consequence that two Bernoulli actions of a group with the properties
in question are Shannon orbit equivalent if and only if they are measure
conjugate. Our arguments apply more generally to actions satisfying a sparse
connectivity condition which we call property SC, and yield an entropy
inequality under the assumption that one of the actions has this property.
