Spanning subgaph with trivial Poisson boundaries

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?)

I don't think much is currently known. Concerning $(\ast\ast)$, there are also finitely presented groups with this property (although they are essentially obtained from wreath products) - see Erschler, Liouville property for groups and manifolds, Invent. Math. 155 (2004). As for $(\ast\ast\ast)$, it is a particular case of the (widely open) question about stability of the Liouville property for finitely generated groups. Namely, whether all simple random walks on a fixed group are Liouville or not simultaneously (i.e., that the Liouville property does not depend on the generating set). In a stronger form this question can be asked about all random walks with a finitely supported non-degenerate symmetric step distribution. It is known that the Liouville property is not stable with respect to rough isometries for simple random walks on graphs or for the Brownian motion on manifolds (which is due to Terry Lyons, 1987), however all known examples are very far from being group invariant.