Maximum monotone paths conjecture for simple 3-dimensional polytopes

Let $P$ be a simple $3$-dimensional polytope with $2n$ vertices, and let $f$ be a generic linear functional on $P$. Let $\b5(P,f)$ denote the number of $f$-monotone paths on $P$, and let $(F_n)$ be the Fibonacci sequence defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 2$. Maximum monotone paths conjecture. We have

$$\mu(P,f)\leq F_{n+2}+1.$$

The proposed maximum can be achieved by wedges of polygons over an edge whose vertices are the source and the sink, with all vertices of the polytope lying on a monotone path; the claim concerns the expected extremal number among simple $3$-dimensional polytopes.