 Solution

Suppose an entry $y$ is bumped from column $j$ of row $i$. If row $i+1$ already contains a cell in column $j$, its entry is greater than $y$ by the strict column condition in the original near Young tableau. When $y$ is inserted into that row, the first entry greater than $y$ therefore occurs in a column at most $j$.

If the next row is shorter than $j$, either a bump occurs earlier or $y$ is appended in column ${\lambda }_{i+1}+1\le j$. This also handles the final appended box. Each successive bumping position is therefore weakly to the left of the previous one. The bumping columns satisfyThis is the row-insertion bumping-path monotonicity needed to preserve the Young-diagram shape and column order.