Staircase heavy-chain conjecture for Grothendieck polynomial weights

Fix $w\in S_n$. A heavy climbing chain of $w$ is a climbing chain together with its marked-link data $M(C)$; write its overlined and ordinary weights as $\operatorname{\overline{wt}}(C,M(C))$ and $\operatorname{wt}(C,M(C))$. Let $C^S(w)$ be the staircase climbing chain and set $\xi=(C^S(w),M(C^S(w)))$. Staircase heavy-chain conjecture. For any term order satisfying $x_1>x_2>\cdots>x_n$,

$$\bm{x}^{\operatorname{\overline{wt}}(\xi)}=\min\left\{\bm{x}^{\operatorname{\overline{wt}}(C,M(C))}\mid C\text{ is a heavy climbing chain of }w\right\}.$$

Equivalently,

$$\bm{x}^{\operatorname{wt}(\xi)}=\max\left\{\bm{x}^{\operatorname{wt}(C,M(C))}\mid C\text{ is a heavy climbing chain of }w\right\}.$$

This conjecture identifies the staircase chain as extremal among heavy climbing chains for the reverse variable order; the source reports verification for $n\leq 8$, while a general proof remains open.