Assume $\Gamma$ is the Cayley graph of an amenable$^{*}$ group and that the simple random walk has non-trivial Poisson boundary$^{**}$. Is there a spanning connected subgraph $\Gamma'$ of some $k$-fuzz$^{***}$ of $\Gamma$ such that the Poisson boundary of $\Gamma'$ is trivial but the growth of balls in $\Gamma'$ is still fast (say faster than any polynomial, and uniformly so at each point)$^{****}$.

$^{*}$though I lack precise arguments in this direction, I believe the answer to this question is "no" if the group is not amenable.

$^{**}$there are results (Kaimanovich) which show (amenable) wreath products such as $\mathbb{Z} \wr \mathbb{Z}$ or $\mathbb{Z}_2 \wr \mathbb{Z}^n$ (with $n \geq 3$) have non-trivial Poisson boundary. Outside wreath products, are there any other amenable groups where this phenomenon is known to occur (for any finitely supported measure)?

$^{***}$a $k$-fuzz of $\Gamma$ is obtained by from $\Gamma$ by adding edges between all points at distance $\leq k$.

$^{****}$ There's a result of B.~Seward which shows amenable groups possess a spanning line (so trivial Poisson boundary). Adding edges to this line seems to be a good idea (say to get a growth of type $e^\sqrt{n}$, but then the resulting graph is no longer a Cayley graph (...and subexponential growth does not guarantee vanishing of the Poisson boundary?)