The infinite-codimension conjecture for products of principal series

Let $K$ be a field, let $P_\alpha$ denote the principal series of order type $\omega^\alpha$, and let $R_\alpha$ denote the corresponding set of products of principal series appearing in the paper. For a $K$-vector space $V$, write $\operatorname{Span}_K(V)$ for its $K$-linear span. Infinite-codimension conjecture. For every ordinal $\alpha>0$,

$$\operatorname{Span}_K(R_\alpha)$$

is infinite-codimensional in

$$\operatorname{Span}_K(P_\alpha)$$

as a $K$-vector space. The paper proves this conclusion for the specified family of ordinals in its theorem, while the stated conjecture extends it to every positive ordinal.