Professor Halfbrain owns a 99×99 board for fantasy chess, whose rows are numbered consecutively from 1 to 99 and whose columns are also numbered consecutively from 1 to 99. A fantasy knight can jump from a square in the 𝑘-th column to any square in the 𝑘-th row (and can jump to no other square); note that if the knight can jump from square 𝑥 to square 𝑦, then this does not mean that it can also jump from square 𝑦 to square 𝑥.
The professor claims that there exists a closed fantasy knight tour on the chessboard that makes the knight visit every square exactly once, and in the end takes it back to its starting square.
Question: Is Halfbrain's claim indeed true, or has the professor once again made one of his mathematical blunders?