McMullen's conjecture on 1-neighborly reorientations

Let $\nu(r,k)$ be the largest $n$ such that every uniform oriented matroid of rank $r$ on $n$ elements has a $k$-neighborly reorientation. McMullen's conjecture. For every $r\geq 3$,

$$\nu(r,1)=2r-1.$$

This conjecture has been verified for $r\leq 5$ but remains open otherwise.