The bounded-support zero-entropy conjecture for polynomial automaton groups

Let $G$ be a polynomial automaton group of degree at most $2$, and let a random walk on $G$ have a step distribution with bounded support. The bounded-support zero-entropy conjecture. Every such random walk has zero asymptotic entropy. This conjecture is open even for degree $d=0$. It concerns whether the symmetry and support assumptions used in the paper are artifacts of the proof, and extends the zero-entropy phenomenon beyond the specially constructed mother-group walks.