Hirschman-Beckner entropy monotonicity conjecture for the block map

Let $n\geqslant 1$, let $f\in L^{\infty}(\mathbb{R}^n)\cap L^{1}(\mathbb{R}^n)\cap L^{2}(\mathbb{R}^n)$, and let $B_{\lambda}$ be the block map. Hirschman-Beckner entropy conjecture. The Hirschman-Beckner entropy decreases under the action of the block map $2^nB_{\lambda}$. The conjecture concerns entropy monotonicity for the real block map; the same question remains open for finite cyclic groups, and the supplied text gives no further resolution information.