Roudneff's conjecture on complete cells in pseudohyperplane arrangements

Let $H(d,n)$ be an arrangement of $n$ pseudohyperplanes in the real projective space $\mathbb{P}^d$, and call a $d$-cell complete if it is bounded by every pseudohyperplane. For an oriented matroid of rank $r$ on $n$ elements, let $C_{r-1}(n)$ denote the number of complete cells in the cyclic arrangement of dimension $r-1$ with $n$ hyperplanes. Roudneff's conjecture. Every rank $r$ oriented matroid $\mathcal{M}$ on $n\geq r+1$ elements has at most

$$2C_{r-1}(n)$$

complete cells. Equivalently, every arrangement of $n\geq 2d+1\geq 5$ pseudohyperplanes in $\mathbb{P}^d$ has at most

$$\sum_{i=0}^{d-2}\binom{n-1}{i}$$

complete cells. The conjecture is known for $d=2,3$, for arrangements arising from Lawrence oriented matroids, and, by this paper's main theorem, for $d=4$.