Regular matching problems for infinite trees

We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(\Sigma\cup\mathcal{X})$ and $R\subseteq T(\Sigma)$ are regular tree languages over finite or infinite trees, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $\sigma_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $\sigma$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $\sigma_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees.