Mondrian partition existence conjecture for squares

Let $S$ be a square, and let $k$ be the number of rectangles in a partition. A proper perfect Mondrian partition is the paper's specified partition of $S$ into $k$ pairwise non-congruent rectangles of equal area. Mondrian partition existence conjecture. Given a square $S$, there is a proper perfect Mondrian partition

$$\mathcal{P}=\{R_1,\dots,R_k\}$$

of $S$ if and only if $k\geq 7$. The paper gives examples and constructions for several values of $k$ and conjectures that the stated threshold is exact for all positive integers $k$.