Cantor-Schr\"oder-Bernstein theorem for a class of countable linear orders

The shuffle of a non-empty countable set $S$ of linear orders is the (unique up to isomorphism) linear order $\Xi(S)$ obtained by fixing a coloring function $\chi: \mathbb{Q} \to S$ having fibers dense in $\mathbb{Q}$ and replacing each rational $q$ in $(\mathbb{Q}, <)$ with an isomorphic copy of $\chi(q)$. We prove that any two countable shuffles that embed as convex subsets into each other are order isomorphic.