Face nonrevisiting conjecture for lattice Hasse diagrams of simple polytopes

Let $P$ be a simple polytope and let ${\bf c}$ be a generic cost vector such that $G(P, {\bf c})$ is the Hasse diagram of a lattice. A directed path in $G(P,{\bf c})$ is said to revisit a face if it leaves that face and later returns to it. Face nonrevisiting conjecture. Directed paths in $G(P, {\bf c})$ never revisit any face they have left. Equivalently, if $v_1\rightarrow v_2\rightarrow \cdots \rightarrow v_k$ is a directed path and $v_1$ and $v_k$ both lie in a face $F$, then $v_i\in F$ for $1\le i\le k$. This property would imply an upper bound of $n-d$ on the length of directed paths, where $d$ is the dimension of $P$ and $n$ is its number of facets. The conjecture proposes a sufficient condition for controlling directed path lengths and is motivated by implications for the diameter of polytope graphs and the efficiency of the simplex method; the supplied text gives no resolution.