Infinite codimension of reducible principal series within principal series of fixed order type

Show that for every ordinal α > 0 and field K of characteristic 0, the K-vector subspace Span_K(R_α) has infinite codimension in Span_K(P_α), where P_α is the set of all principal series b ∈ K((ℝ^{≤0})) with ot(b) = ω^α and sup(supp(b)) = 0, and R_α ⊆ P_α is the set of reducible elements.

Background

For many specific forms of α, the authors prove that the space spanned by reducible principal series (of order type ω^α) is an infinite-codimension subspace of the space spanned by all principal series of that order type. This provides a strong sense in which “most” principal series of fixed order type are irreducible.

They conjecture that this infinite codimension phenomenon holds uniformly for every α > 0, extending their proven cases beyond the families covered in their main theorems.