Variant of KBB3 Problem 4

This is a harder version of KBB3 Problem 4.

Suppose $n$ is an odd positive integer. Given an $n \times n$ chessboard where each cell is colored black or white. At any step, we are allowed to pick any cell and flip the colors of the cells that are in the same row or the same column as that cell. Prove or disprove: no matter what the initial coloring is, we can always get an all-white board?

KBB3 Problem 4

Given a 3x3 chessboard where each cell is colored black or white. At any step, we are allowed to pick any column or any row and flip all the cell colors in that column or row. Prove or disprove: no matter what the initial coloring is, we can always get an all-white board?