Strict monotone Hirsch conjecture for polytope orientations

Let $P$ be a $d$-dimensional polytope with $n$ facets, and let ${\bf c}$ be a generic linear functional. Orient the graph of $P$ according to increasing values of ${\bf c}$, and denote this directed graph by $G(P,{\bf c})$. Its unique minimum and maximum vertices are called the source and sink, respectively. Strict Monotone Hirsch Conjecture. There is a directed path in $G(P,{\bf c})$ from the source of $P$ to the sink of $P$ of length at most $n-d$. The conjecture concerns a monotone analogue of the Hirsch bound and is presented as a stronger condition related to the face nonrevisiting question; the supplied text gives no resolution.