Cantor-Schröder-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 $\mathrm{\Xi }(S)$ 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.