Hafner's 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. The Rajchgot index $\mathrm{raj}(w)$ is the degree of $\mathfrak{G}_w$, and $\bm{x}^{\operatorname{rajcode}(w)}$ is the leading monomial of its highest-degree homogeneous component. Hafner's conjecture. For $\ell(w)<k\leq \mathrm{raj}(w)$,

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

where $p$ is the largest index such that $x_p m_{k-1}(\bm{x})$ divides $\bm{x}^{\operatorname{rajcode}(w)}$. This conjecture gives a recursive description of the leading monomials of all positive-degree homogeneous components of a Grothendieck polynomial; its status is not specified in the source.