Hafner's reverse-order recursive leading-monomial conjecture for Grothendieck polynomials

Fix $w\in S_n$ and any term order with $x_1>x_2>\cdots>x_n$. Let $\mathfrak{G}_w^{(k)}$ denote the degree-$k$ homogeneous component of $\mathfrak{G}_w$, and let $m_k(\bm{x})$ be its leading monomial. Let $C^S(w)$ be the staircase climbing chain with marked-link data $M(C^S(w))$, and write $\operatorname{wt}(C^S(w),M(C^S(w)))$ for its weight. Hafner's reverse-order conjecture. For $\ell(w)<k\leq \mathrm{raj}(w)$,

$$m_k(\bm{x})=x_p m_{k-1}(\bm{x}),$$

where $p$ is the smallest index such that $x_p m_{k-1}(\bm{x})$ divides $\bm{x}^{\operatorname{wt}(C^S(w),M(C^S(w)))}$. The conjecture predicts the leading monomials of the homogeneous components in the reverse variable order, using the staircase chain; it is presented as an analogue of the preceding conjecture and remains open in the supplied source.