Quantum central limit 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. Quantum central limit conjecture. Under iteration of the block map $2^nB_{\lambda}$, $f$ converges either to $0$ or to a Gaussian function. This is presented as a quantum analogue of a central limit phenomenon, following the discussion of biprojections as limit points in the finite-index irreducible subfactor setting; the supplied text gives no resolution status.