  The Holt-Klee theorem says that the graph of a $d$-polytope, with edges
oriented by a linear function on $P$ that is not constant on any edge, admits
$d$ independent monotone paths from the source to the sink. We prove that the
digraphs obtained from oriented matroid programs of rank $d+1$ on $n+2$
elements, which include those from $d$-polytopes with $n$ facets, admit $d$
independent monotone paths from source to sink if $d \le 4$. This was
previously only known to hold for $d\le 3$ and $n\le 6$.
