  Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of the
complete graph $K_n$ in the plane (equivalently, the sphere) is
$\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor=n^4/64
+ O(n^3)$. Moon proved that the expected number of crossings in a spherical
drawing in which the points are randomly distributed and joined by geodesics is
precisely $n^4/64+O(n^3)$, thus matching asymptotically the conjectured value
of $\text{cr}(K_n)$. Let $\text{cr}_P(G)$ denote the crossing number of a graph
$G$ in the projective plane. Recently, Elkies proved that the expected number
of crossings in a naturally defined random projective plane drawing of $K_n$ is
$(n^4/8\pi^2)+O(n^3)$. In analogy with the relation of Moon's result to Hill's
conjecture, Elkies asked if $\lim_{n\to\infty} \text{cr}_P(K_n)/n^4=1/8\pi^2$.
We construct drawings of $K_n$ in the projective plane that disprove this.
