Mészáros–Setiabrata–St. Dizier's unit-step support conjecture for Grothendieck polynomials

Fix $w\in S_n$ and write

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

For a monomial exponent $\alpha$, write $|\alpha|=\alpha_1+\cdots+\alpha_n$, and order exponents componentwise: $\alpha<\beta$ when $\alpha_i\leq\beta_i$ for every $i$ and $\alpha\ne\beta$. Let $\operatorname{supp}(\mathfrak{G}_w(x))$ be the set of exponents with nonzero coefficient. Mészáros–Setiabrata–St. Dizier's unit-step support conjecture. If $\alpha\in\operatorname{supp}(\mathfrak{G}_w(x))$ and $|\alpha|<\deg(\mathfrak{G}_w(x))$, then there exists $\beta\in\operatorname{supp}(\mathfrak{G}_w(x))$ with $\alpha<\beta$ and $|\beta|=|\alpha|+1$. This is a natural strengthening of the preceding support-extension property; the paper proves related conjectures in the zero-one case, but the general assertion remains open.