$k$-Roudneff's conjecture for neighborly reorientations of oriented matroids

Let $\mathcal{M}$ be a rank-$r$ oriented matroid on $n$ elements, and let $f_{\mathcal{M}}(r,n,k)$ denote the number of $k$-neighborly reorientations of $\mathcal{M}$. Let $C_r(n)$ be the rank-$r$ alternating oriented matroid on $n$ elements, and set $c_r(n,k)=f_{C_r(n)}(r,n,k)$.

$k$-Roudneff's conjecture. For any rank $r\geq 2k+1\geq 3$ oriented matroid $\mathcal{M}$ on $n\geq r+1$ elements,

$$f_{\mathcal{M}}(r,n,k)\leq c_r(n,k).$$

The conjecture holds for $k=2$, $r=6$, for every odd rank $r=2k+1$, and in several other cases, including $k=1$ with $r\leq 5$ and Lawrence oriented matroids. It remains open in general.