Infinite codimension of reducible principal series for all ordinals

Show that for every ordinal α > 0, the K-linear span Span_K(R_α) of reducible principal series of order type ω^α (i.e., elements of P_α that factor nontrivially in K((R^{≤0}))) has infinite codimension in the K-vector space Span_K(P_α) spanned by all principal series P_α (those b with ot(b) = ω^α and sup(supp(b)) = 0).

Background

The authors define P_α as the set of principal series with order type ω^α and supremum of support equal to 0, and R_α as the subset of P_α consisting of reducible elements. For several families of α handled by their main theorems, they prove that the K-span of reducible principal series is of infinite codimension inside the K-span of all principal series of that order type.

They conjecture that this infinite codimension phenomenon holds uniformly for every α > 0, thereby strengthening their partial results to a complete picture across all ordinals.