Conjecture that the two box-move algorithms produce the same sequence

Let $X$ and $Z$ be LR-fillings of the same type, with the skew shape $\beta\setminus\gamma$ a rook strip and $Z<_{\sf dom}X$. The first and second algorithms produce LR-fillings $Y_1$ and $Y_2$ satisfying $Z\leq_{\sf dom}Y_i$ and $Y_i<_{\sf box}X$ for $i=1,2$. In the second algorithm, choose the position $\ell$ of the second entry as large as possible; in the first algorithm, use the reduced expression obtained by the specified bubble-sort factorization. Conjecture that the two algorithms produce the same result. With these choices, $Y_1=Y_2$. This open problem compares two procedures for moving between LR-fillings in the dominance and box orders; the source provides no resolution.