Proof of a K-theoretic polynomial conjecture of Monical, Pechenik, and Searles

As part of a program to develop K-theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas ${\bar{A}}_a={\sum }_b{Q}_b^a\left(\beta \right){\bar{P}}_b$ and ${\bar{Q}}_a={\sum }_b{M}_b^a\left(\beta \right){\bar{F}}_b$, where each of ${\bar{A}}_a$, ${\bar{P}}_a$, ${\bar{Q}}_a$ and ${\bar{F}}_a$ is a family of polynomials that forms a basis for $Z[x_1,\dots ,x_n]\left[\beta \right]$ indexed by weak compositions a, and $Q_b^a\left(\beta \right)$ and $M_b^a\left(\beta \right)$ are monomials in β for each pair $(a,b)$ of weak compositions. The polynomials ${\bar{A}}_a$ are the Lascoux atoms, ${\bar{P}}_a$ are the kaons, ${\bar{Q}}_a$ are the quasiLascoux polynomials, and ${\bar{F}}_a$ are the glide polynomials; these are respectively the K-analogues of the Demazure atoms $A_a$, the fundamental particles $P_a$, the quasikey polynomials $Q_a$, and the fundamental slide polynomials $F_a$. Monical, Pechenik, and Searles conjectubolded that for any fixed a, ${\sum }_b{Q}_b^a(-1),{\sum }_b{M}_b^a(-1)\in \{0,1\}$, where b ranges over all weak compositions. We prove this conjecture using a sign-reversing involution.