The quotient conjecture for Lie algebras of self-similar groups

Let $G$ be a self-similar group as described in the source, and let ${\mathscr L}^\Bbbk(G)$ be its associated Lie algebra over the field $\Bbbk$. Let ${\mathscr L}(G)$ be the self-similar Lie algebra constructed from $G$. Quotient conjecture. Then

$${\mathscr L}(G)\text{ is a quotient of }{\mathscr L}^\Bbbk(G).$$

The assertion relates the self-similar construction to the Lie algebra associated with the dimension-series or augmentation-filtration construction; the supplied text does not indicate whether it has been proved or disproved.