MathNet 2022-part-01 — 197 problems (EN–VI) Question 54

The cells of the $2022 \times 2022$ table are filled with positive integers from $1$ to $2022^2$, each cell contains exactly one number, all numbers are used exactly once. For each line Vlad wrote out one number which is the second in descending order in this line. And Dima did the same for each column. It turned out that the boys wrote down $4044$ pairwise distinct numbers and there are exactly $k$ numbers written down by Vlad, such that each of them is less than every number written down by Dima.
Find the largest possible value of $k$. (Mikhail Karpuk)