A Clever Combinatorics Puzzle from the 2023 USA Math Olympiad

Problem 

Let $n ≥ 3$ be an integer. We say that an arrangement of the numbers $1, 2, …, n²$ in an $n × n$ table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression.

For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

See solution here.