Solution
= Solution
\b[False in general.] On $[5]$ take the rank-two star
$$
\mathcal A=\{12,13,14,15\}.
$$
It is an <intersecting family> of four sets and $r=2\le5/2$. The first four rank-two sets in <colexicographic order> are
$$
\{12,13,23,14\}.
$$
The members $23$ and $14$ are disjoint. Thus <colexicographic replacement can destroy intersection>, despite its shadow-minimizing property.