Feng–Huang–Rao's algebraic dependence conjecture for affine embeddings of Cantor sets

Let $F$ and $E$ be totally disconnected self-similar sets in $\mathbb{R}^d$, generated by the iterated function systems $\Phi=\lbrace\phi_i\rbrace_{i=1}^l$ and $\Psi=\lbrace\psi_j\rbrace_{j=1}^m$, respectively. Let $\alpha_i\in(0,1)$ and $\beta_j\in(0,1)$ denote the contraction ratios of $\phi_i$ and $\psi_j$. An affine embedding of $F$ into $E$ is an affine map whose restriction maps $F$ into $E$. Feng–Huang–Rao's conjecture. If $F$ can be affinely embedded into $E$, then for every $1\leq i\leq l$ there exist $t_{i,j}\in\mathbb{Q}$ with $t_{i,j}\geq0$ such that

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

In particular, if $\beta_j=\beta$ for all $1\leq j\leq m$, then for every $1\leq i\leq l$,

$$\frac{\log\alpha_i}{\log\beta}\in\mathbb{Q}.$$

The conjecture predicts arithmetic restrictions on contraction ratios whenever one totally disconnected self-similar set affinely embeds into another. Its status is not resolved by the supplied source context.