Feng–Hochman–Rogers contraction-ratio conjecture for affine embeddings

Let $E,F\subset\mathbb{R}^d$ be two totally disconnected non-trivial self-similar sets, generated by iterated function systems $\Phi=\{\varphi_i\}_{i\in I}$ and $\Psi=\{\psi_j\}_{j\in J}$, respectively. Let $r_i$ and $r_j'$ denote the contraction ratios of $\varphi_i$ and $\psi_j$, respectively. Suppose that $F$ can be affinely embedded into $E$. Feng–Hochman–Rogers conjecture. For each $j\in J$, there exist non-negative rational numbers $t_{i,j}$ such that

$$r_j'=\prod_{i\in I}r_i^{t_{i,j}}.$$

In particular, if $r_i=r$ for all $i\in I$, then

$$\frac{\log r_j'}{\log r}\in\mathbb{Q}$$

for all $j\in J$. This conjecture concerns the arithmetic dependence forced by affine embeddings between totally disconnected self-similar sets; the source provides no evidence of resolution.