The affine-embedding conjecture for self-similar sets

Let $\Phi=\{\phi_i(x)=\alpha_i x+a_i\}_{i=1}^m$ and $\Psi=\{\psi_j(x)=\beta_jx+b_j\}_{j=1}^{\ell}$ be self-similar iterated function systems on $\mathbb R$, with attractors $X_\Phi$ and $X_\Psi$, respectively. The strong separation condition (SSC) means that the basic pieces of the IFS are pairwise disjoint. The affine-embedding conjecture. Assume that $X_\Psi$ is not a singleton, that $\Phi$ satisfies the SSC, and that $\dim_{\rm H}X_\Phi<1$. If there are real numbers $u\ne0$ and $v$ such that $uX_\Psi+v\subset X_\Phi$, then for every $1\le j\le\ell$ there are rational numbers $r_{i,j}\ge0$ such that

$$\beta_j=\prod_{i=1}^m\alpha_i^{r_{i,j}}.$$

The source describes this as a special case of a conjecture in the cited work and does not provide a resolution in the supplied passage.