An S-tileset is a collection of n oriented tiles, where no two tiles have the same size, each tile is one unit thick, and its non-zero-integer length and width add up to n+1. (So, an S-tileset has n tiles, of sizes 1×n, 2×n−1, ...n×1.).
Conjecture: No two disjoint subsets of any S-tileset can be tiled to form two areas of identical size and shape.
Can you either prove the conjecture or find a counter-example.