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 $\overline{\mathfrak{A}}_a = \sum_b Q_b^a(\beta)\overline{\mathfrak{P}}_b$ and $\overline{\mathfrak{Q}}_a = \sum_b M_b^a(\beta)\overline{\mathfrak{F}}_b,$ where each of $\overline{\mathfrak{A}}_a$, $\overline{\mathfrak{P}}_a$, $\overline{\mathfrak{Q}}_a$ and $\overline{\mathfrak{F}}_a$ is a family of polynomials that forms a basis for $\mathbb{Z}[x_1,\dots,x_n][\beta]$ indexed by weak compositions $a,$ and $Q_b^a(\beta)$ and $M_b^a(\beta)$ are monomials in $\beta$ for each pair $(a,b)$ of weak compositions. The polynomials $\overline{\mathfrak{A}}_a$ are the Lascoux atoms, $\overline{\mathfrak{P}}_a$ are the kaons, $\overline{\mathfrak{Q}}_a$ are the quasiLascoux polynomials, and $\overline{\mathfrak{F}}_a$ are the glide polynomials; these are respectively the $K$-analogues of the Demazure atoms $\mathfrak{A}_a$, the fundamental particles $\mathfrak{P}_a$, the quasikey polynomials $\mathfrak{Q}_a$, and the fundamental slide polynomials $\mathfrak{F}_a$. Monical, Pechenik, and Searles conjectured 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.