Siegel's conjecture on intersections of Hirsch polytopes with cubes

Let $H$ be a polytope satisfying the Hirsch Conjecture, and let $C$ be a cube. The intersection $H\cap C$ is again a polytope.

Siegel's Conjecture. If $H$ satisfies the Hirsch Conjecture and $C$ is a cube, then $H\cap C$ satisfies the Hirsch Conjecture.

The conjecture was proposed as a statement about preserving the Hirsch property when a polytope is intersected with a cube, motivated by restricting every decision variable in a linear program to lie between $0$ and $1$. The paper constructs counterexamples to this conjecture, so it is refuted.