Existence of irreducible principal series of order type ω^{3ω+2}

Determine whether there exists any irreducible principal series b in K((R^{≤0})) with order type ω^{3ω+2} and sup(supp(b)) = 0; equivalently, decide whether P_{3ω+2} contains an irreducible element.

Background

Within their inductive analysis of principal elements by Cantor normal form, the authors establish irreducibility for many cases, including α = 3ω + 1. However, the next case α = 3ω + 2 remains unsettled.

They explicitly note that α = 3ω + 2 is currently the smallest ordinal for which they do not know if any irreducible element exists in P_α (the set of principal series of order type ω^α).