Mészáros–Setiabrata–St. Dizier's coefficient-sum conjecture for Grothendieck polynomials

Fix $w\in S_n$ and write

$$\mathfrak{G}_w(x)=\sum_{\alpha\in\mathbb{Z}^n} C_{w\alpha}x^\alpha.$$

Let $\operatorname{supp}(\mathfrak{G}_w^{\mathrm{top}}(x))$ denote the exponents of monomials of highest degree in $\mathfrak{G}_w(x)$, and order exponents componentwise. Mészáros–Setiabrata–St. Dizier's coefficient-sum conjecture. For any $\beta\in\operatorname{supp}(\mathfrak{G}_w^{\mathrm{top}}(x))$, one has

$$\sum_{\alpha\leq\beta} C_{w\alpha}=1.$$

This conjecture concerns the coefficients below each top-degree monomial and is part of the support and coefficient properties studied in the paper; its general resolution is not supplied in the stated context.