Bottom-tree invariance of rectification in d-complete posets

Let $\mathcal{P}$ be a $d$-complete poset with bottom tree $\mathcal{B}$. Let $T$ be a skew increasing $\mathcal{P}$-tableau, and let $R$ and $S$ be rectifications of $T$. Bottom-tree rectification conjecture. The restrictions of the rectifications to the bottom tree agree:

$$R|_{\mathcal{B}}=S|_{\mathcal{B}}.$$

This generalizes the proved unique-rectification result for chained double-tailed diamond posets and provides additional experimental evidence toward the broader unique-rectification-target conjecture. It asserts invariance on the bottom tree even when the full rectification need not yet be known to be unique.