Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a transitive sofic subshift such that $h_{top}(\sigma_X) < h_{top}(\Sigma_{Y})$.

My question is: What are the conditions (if any) to be sure that there is a proper embedding from X to Y?

It is known that if $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ are both transitive subshifts of finite type then such embedding exists if and only if $h_{top}(\sigma_X) < h_{top}(\sigma_{Y})$ and $Per_n(\sigma_X) \leq Per_n(\sigma_Y)$ for every $n \in \mathbb{N}$ where $Per_n$ denotes the cardinality of the set of points having least period $n$. I believe that $h_{top}(\sigma_X) < h_{top}(\Sigma_{Y})$ is a necessary condition. Also, the examples that I am working with satisfy that $Per(\sigma_X) \cap Per(\sigma_Y) = \emptyset$.