The Lee (2026) conjecture on the diameters of order and chain polytopes fails in both directions

For a finite poset P, Stanley's order polytope 𝒪(P) and chain polytope 𝒞(P) have the same number of vertices and, by a theorem of Hibi, Li, Sahara and Shikama, the same number of edges. In 2026 Lee conjectured that for a connected poset the diameters of the graphs of these two polytopes differ by at most 1. We show that this fails in both directions. For every k ≥ 1 we describe a connected poset Bk with 3k elements whose order polytope has graph diameter k and whose chain polytope has graph diameter at most ⌊3(k+1)/4⌋; hence the first diameter can exceed the second by any amount, and by at least 2 already for k = 8. For every n ≥ 4 we construct a connected poset Pn with 2n2 + 5n + 1 elements whose order polytope has graph diameter 2 and whose chain polytope has graph diameter 4. Finally, for every finite poset P, if the graph of 𝒪(P) has diameter at most 2, then the graph of 𝒞(P) has diameter at most 4; so 2 is the largest possible difference when the order polytope has graph diameter 2.