= 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 $\lambda_{i+1}+1\leq j$. This also handles the final appended box. Each successive bumping position is therefore weakly to the left of the previous one. \b[The bumping columns satisfy]
$$
\boxed{j'\geq j''\geq j'''\geq\cdots}.
$$
This is the <row-insertion bumping-path monotonicity> needed to preserve the Young-diagram shape and column order.
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