The subalgebra conjecture for thinned self-similar group algebras

Let $G$ be a self-similar group as described in the source, let ${\mathscr A}(G)$ be its thinned algebra with augmentation ideal $\varpi$, and define its associated graded algebra by

$$\overline{{\mathscr A}(G)}=\bigoplus_{n\ge0}\varpi^n/\varpi^{n+1}.$$

Let ${\mathscr L}(G)$ be the associated self-similar Lie algebra. Subalgebra conjecture. Then ${\mathscr L}(G)$ is a subalgebra of $\overline{{\mathscr A}(G)}$. The claim places the self-similar Lie algebra inside the graded algebra arising from the thinned group algebra; the supplied text does not indicate whether it has been proved or disproved.