The mother-group zero-entropy random-walk conjecture

Let $d$ be the degree of a mother group, and consider random walks whose step distribution is supported on a finite generating set. The mother-group zero-entropy conjecture. The mother group of degree $d$ has such a random walk with asymptotic entropy zero if and only if $d=0,1,2$. The case $d=1$ follows from the paper, while the assertion for all $d\geq2$ remains open. It predicts a phase transition in the existence of zero-asymptotic-entropy random walks on mother groups.