Polynomial bound for the monotone diameter of lattice polytopes

The conjecture asks whether monotone diameters of lattice polytopes admit a polynomial bound in the dimension and the box parameter $k$. The reported counterexample at $k=3$ would refute the general claim and separates the known linear cases $k=1,2$ from the failure at $k=3$.

Known results

• Half-integral polytopes have monotone diameter at most $3d$; the general question was explicitly open in 2022.
• For $d$-dimensional $(m+1)$-level polytopes, the bound is $(d-1)m+1$.
• Under stated integral-description assumptions, a bound of $d^2k\lVert A\rVert_\infty$ is known.

Exponential counterexample at $k=3$

A new arXiv preprint, Monotone Diameters of Lattice Polytopes, claims exponential monotone diameter for $k=3$, together with related examples for unbounded polyhedra. If correct, this settles the proposed universal polynomial bound negatively; the preprint is unrefereed.

Current status (as of September 2026): The cases $k=1,2$ have known linear bounds, while the general polynomial-bound conjecture is claimed false at $k=3$; the counterexample remains unverified.