Unique rectification targets in d-complete posets

Let $\mathcal{P}$ be a $d$-complete poset and let $\lambda \subseteq \mathcal{P}$ be an order ideal. A rectification target is an increasing $\mathcal{P}$-tableau obtained as the result of rectifying a skew increasing tableau. Unique rectification target conjecture. There is an explicitly defined unique rectification target supported on $\lambda$. This conjecture asks whether general $d$-complete posets have enough unique rectification targets to extend the Thomas–Yong jeu de taquin theory beyond minuscule varieties; the precise formulation identifies the target with the minimally labeled tableau described below.