On an Extremal Data Processing Inequality for long Markov Chains

We pose the following extremal conjecture: Let X,Y be jointly Gaussian random variables with linear correlation ρ. For any random variables U, V for which U,X, Y, V form a Markov chain, in that order, we conjecture that: 2−2[I(X;V )+I(Y ;U)] ≥ (1− ρ2)2−2I(U ;V ) + ρ22−2[I(X;U)+I(Y ;V . By letting V be constant, we see that this inequality generalizes a well-known extremal result proved by Oohama in his work on the quadratic Gaussian one-helper problem. If valid, the conjecture would have some interesting consequences. For example, the converse for the quadratic Gaussian two-encoder source coding problem would follow from the converse for multiterminal source coding under logarithmic loss, thus unifying the two results under a common framework. Although the conjecture remains open, we discuss both analytical and numerical evidence supporting its validity.