Derivation of the modified q,t-Kostka stability range from plethystic formulas

Prove Corollary \ref{qtKostkaRange} directly from the plethystic formulas for the modified \(q,t\)-Kostka numbers in equations (\ref{plethForm}) or (\ref{k_lambda}), while controlling the negative powers of \(q\) and \(t\) introduced by those formulas.

Background

The paper proves a stable range for coefficients of modified Macdonald polynomials by counting admissible labelings and constructing stabilization bijections. It then records two alternative plethystic descriptions of the modified $q,t$-Kostka numbers. Because these descriptions involve Laurent polynomials and the operator $\nabla^{-1}$ can introduce negative powers, the argument used for hook shapes does not directly extend. The authors explicitly leave open whether a careful analysis of either formula can yield the same stability result.