Feng–Huang–Rao affine embedding conjecture for self-similar sets

Let $X,Y\subseteq\mathbb{R}$ be non-trivial strongly separated self-similar sets generated by non-constant contracting affine maps $(\varphi_i)_{i\in I}$ and $(\psi_j)_{j\in J}$, respectively. Let $\alpha_i=\left\Vert\varphi_i\right\Vert$ and $\beta_j=\left\Vert\psi_j\right\Vert$ be their scaling constants, let $\mathcal{A}$ be the set of non-singular affine maps $\mathbb{R}\to\mathbb{R}$, and define the set of affine embeddings of $X$ into $Y$ by

$$\mathcal{E}=\\{f\in\mathcal{A}:f(X)\subseteq Y\\}.$$

Feng–Huang–Rao's conjecture. If, for some $i\in I$, $\log\alpha_i\notin\operatorname{span}_{\mathbb{Q}}\\{\log\beta_j\\}_{j\in J}$, then

$$\mathcal{E}=\emptyset.$$

The conjecture concerns the arithmetic obstruction to affine embeddings between strongly separated self-similar sets. The paper proves it for Lebesgue-almost all scaling parameters and establishes several additional cases, but the supplied source does not indicate that the full conjecture has been resolved.