Implication of refined Macdonald positivity for the original LLM k-Schur conjecture

Determine whether the refined Macdonald positivity result stated in Corollary \ref{cor:rMac}, formulated using the paper’s definition of k-Schur functions, implies the original refined Macdonald positivity conjecture of Lapointe, Lascoux, and Morse.

Background

The paper proves a refined Macdonald positivity statement for a definition of k-Schur functions that agrees with the framework used in several later works, including Chen–Haiman and Blasiak–Morse–Pun–Summers. The authors note that this definition differs from the one used in the original conjecture of Lapointe, Lascoux, and Morse.

Consequently, it remains unresolved whether the positivity assertion established as Corollary \ref{cor:rMac} is strong enough to imply the original conjecture. Resolving this would clarify the relationship between the paper’s categorified positivity theorem and the historically original formulation of refined Macdonald positivity.