Shelah's five-element basis conjecture for uncountable linear orders

Let $X$ be a set of reals of cardinality $\aleph_1$, let $\omega_1$ be the first uncountable ordinal, let $\omega_1^*$ be its converse order, and let $C$ be a Countryman line, meaning an uncountable linear order whose square is the union of countably many chains under the coordinate-wise partial order. Write $C^*$ for the converse of $C$. Shelah's five-element basis conjecture. Under the Proper Forcing Axiom, the orders $X$, $\omega_1$, $\omega_1^*$, $C$, and $C^*$ form a five-element basis for the uncountable linear orders whenever $X$ is a set of reals of cardinality $\aleph_1$ and $C$ is a Countryman line. The conjecture concerns the classification of uncountable linear orders by embeddability; the source explains that Countryman lines exist in ZFC and that the five-element basis was viewed as the natural formulation after the development of proper forcing and the Proper Forcing Axiom.