The unresolved exponent pattern in a linear-polynomial equivalence

Determine whether there exists a Ramsey pair (α, β) such that nα + mβ is equivalent to kα⁻¹β, namely the case (r, s) = (−1, 1) in the relation nα + mβ ∼ kαʳβˢ.

Background

The paper classifies several relations of the form nα + mβ ∼ kαʳβˢ for Ramsey pairs, reducing the remaining possibilities to a small set of exponent pairs. The cases (0, 1) and (1, 0) are handled using the preceding polynomial results, while the case (−1, 1), corresponding to kβ/α, is not resolved except for consequences already implied by Corollary 5.12.