Bader–Muchnik conjecture on $L^2$-irreducibility of Poisson boundaries

Let $(\Gamma,\tau)$ be a measured group, and let its Poisson boundary be $(B,\nu)$. For $p\in[1,\infty)$, the associated representation is

$$\sigma_p(\gamma)f=\left(\frac{d\gamma\nu}{d\nu}\right)^{1/p}f(\gamma^{-1}\cdot),$$

for $\gamma\in\Gamma$ and $f\in L^p(B,\nu)$. The boundary is $L^p$-irreducible if, for every nonzero $f\in L^p(B,\nu)$, the linear span of $\{\sigma_p(\gamma)f\mid\gamma\in\Gamma\}$ is norm-dense in $L^p(B,\nu)$. Bader–Muchnik conjecture. The Poisson boundary of a measured group is $L^2$-irreducible. This conjecture concerns the density of orbit spans in the natural boundary representation and is presented here as an influential conjecture; the supplied source does not state a resolution.