The commuting infinite subgroups conjecture for intermediate-growth groups

Let $G$ be a group of intermediate growth. Two subgroups commute with each other when $[h_1,h_2]=1$ for every $h_1\in H_1$ and $h_2\in H_2$.

Commuting infinite subgroups conjecture. The group $G$ contains two infinite subgroups $H_1$ and $H_2$ which commute with each other:

$$[h_1,h_2]=1$$

for all $h_1\in H_1$ and $h_2\in H_2$.

The source presents this as a final, technical conjecture and notes that, if true, it would positively resolve the $p_c<1$ conjecture of Benjamini and Schramm for percolation on Cayley graphs.