Alternating dual Pieri rule for closed k-Schur Katalan functions

Prove the alternating dual Pieri rule for every k-bounded partition λ and every nonnegative integer m: if the coefficients c^λ_μ are defined by G^{\perp}_{1^m}\,\widetilde g^{(k)}_λ=\sum_{μ\in P}c^λ_μ\,\widetilde g^{(k)}_μ, then establish that (−1)^{|λ|−|μ|−m}c^λ_μ\in\mathbb Z_{\ge 0} for every partition μ.

Background

Closed k-Schur Katalan functions \widetilde g^{(k)}_λ form a linear basis of the symmetric-function algebra \Lambda^{(k)}. Blasiak, Morse, and Seelinger proposed an alternating dual Pieri rule governing the coefficients obtained by applying the adjoint G^{\perp}_{1^m} of a stable Grothendieck polynomial to a closed k-Schur Katalan function.

The paper proves this sign-positivity statement only when k is sufficiently large and when λ is a strictly decreasing partition. The unrestricted assertion for arbitrary k-bounded partitions is therefore an explicitly stated conjecture that remains unresolved in the paper.