We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the $d$-step conjecture of Klee and Walkup in the case $d = 6$. This implies that for all pairs $(d, n)$ with $n − d ≤ 6$, the diameter of the edge graph of a $d$-polytope with $n$ facets is bounded by 6, which proves the Hirsch conjecture for all $n − d ≤ 6$. We prove this result by establishing this bound for a more general structure, so-called matroid polytopes, by reduction to a small number of satisfiability problems.
"Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets." Experiment. Math. 20 (3) 229 - 237, 2011.