The $K$-theoretic coefficient-sum conjecture for glides and kaons

A weak composition is a finite sequence of nonnegative integers. For weak compositions $a$ and $b$, let $M^a_b(\beta)$ be the coefficient of the glide polynomial $\overline{\mathfrak{F}}_b$ in the glide expansion of the quasiLascoux polynomial $\overline{\mathfrak{Q}}_a$, and let $Q^a_b(\beta)$ be the coefficient of the kaon $\overline{\mathfrak{P}}_b$ in the kaon expansion of the Lascoux atom $\overline{\mathfrak{A}}_a$. Both are nonnegative monomials in $\beta$. The coefficient-sum conjecture. For every weak composition $a$,

$$\sum_b M^a_b(-1)\in\{0,1\}\quad\text{and}\quad\sum_b Q^a_b(-1)\in\{0,1\},$$

where both sums range over all weak compositions $b$. These conjectures seek geometric interpretations of the relevant $K$-theoretic polynomials and of the coefficient sums as Euler characteristics; no cohomological analogue is expected.