Satisfiability of String Constraints with Subsequence relation

By $\le$ we denote the (scattered) subsequence relation between words. Example: $aba \le baabbabbb$. Fix a finite alphabet $A$. Let $X$ be a set of string variables. A subsequence constraint is an expression of the form $x \le \alpha$ where $x \in X$ and $\alpha \in X^*$. A homomorphism $h: X^\ast \to A^*$ satisfies a subsequence constraint $x \le \alpha$ if $h(x) \le h(\alpha)$. A domain restriction constraint $d: X \to RE(A)$ restricts the domain of possible values for a variable $x$ to the regular language given by the regular expression $d(e)$. We are interested in the decidability of the following problem: Input: A set $C$ of subsequence constraints and a domain restriction $d$. Question:Is there a homomorphism $h$ that respects the domain restriction $d$ and satisfies every subsequence constraint in $C$?