Can all large integer $n \times(n+1)$ rectangles be tiled with smaller copies of distinct $k\times(k+1)$ rectangles?

Question

Can all large integer $n\times(n+1)$ rectangles be tiled with smaller copies of distinct $k\times(k+1)$ rectangles? It seems to be true for $n\geq20$, but I don't see how to do it in general.

https://erich-friedman.github.io/mathmagic/0512.html