Data processing inequality for interaction information

Question

The interaction information is defined as $I(X;Y)-I(X;Y|Z)$. Let $Z-(X, Y) -(X', Y')$ be a Markov chain. Is there an inequality similar to the data processing inequality, relating $I(X';Y')-I(X';Y'|Z)$ to $I(X;Y)-I(X;Y|Z)$? Thanks in advance.

Answer 1

Since the interaction information can be either negative or positive, and a Markov chain that erases everything can be used to take the interaction information to $0$, without further conditions the answer is "no".