Sublinear growth of Lipschitz harmonic functions with trivial Poisson boundary

Let $G$ be an amenable group, let $\mu$ be a probability measure on $G$ with finite second moment and trivial Poisson boundary, and let $u:G\to\mathbb R$ be a Lipschitz $\mu$-harmonic function. Assume that

$$\int \frac{1}{|g|}|u(g)|\,d\mu_n(g)\to 0$$

for every Reiter sequence $(\mu_n)$. The sublinear-growth conjecture. The function $u$ has sublinear growth. This is posed as an open problem in the paper; the hypotheses connect harmonic-function growth with averaging over Reiter sequences, and no resolution is supplied here.