Robichaux's ghost Kohnert rule for Grothendieck polynomials

For a permutation $w\in S_n$, let ${\sf GKKoh}(w)$ be the family of ghost $K$-Kohnert diagrams generated from the diagram of $w$ by ordinary, $K$-, ghost, and ghost $K$-Kohnert moves. For $\alpha\in\mathbb{Z}_{\geq 0}^n$, let ${\sf GKKoh}(w,\alpha)$ consist of the diagrams of weight $\alpha$, and let $\mathrm{PD}(w,\alpha)$ be the pipe dreams of $w$ of weight $\alpha$.

Robichaux's ghost Kohnert conjecture. For every permutation $w\in S_n$ and $\alpha\in\mathbb{Z}_{\geq 0}^n$,

$$\#{\sf GKKoh}(w,\alpha)=\#\mathrm{PD}(w,\alpha).$$

This proposed correction to the Ross–Yong rule is the conjectured full enumerative model for Grothendieck polynomials. The supplied text gives no resolution, while the paper's results motivate the weaker support conjecture because the full rule may fail by overcounting and undercounting.