Hill's crossing-number conjecture for complete graphs

Let $K_n$ be the complete graph on $n$ vertices, and let $\operatorname{cr}(K_n)$ denote its crossing number, the minimum number of crossings in a drawing of $K_n$ in the plane or sphere. Define

$$X(n):=\frac{1}{4}\Big\lfloor \frac{n}{2}\Big\rfloor \Big\lfloor \frac{n-1}{2} \Big\rfloor \Big\lfloor \frac{n-2}{2} \Big\rfloor \Big\lfloor \frac{n-3}{2}\Big\rfloor.$$

Hill's conjecture. The crossing number satisfies

$$\operatorname{cr}(K_n)=X(n)=\frac{3}{8}\binom{n}{4}+O(n^3).$$

This is a foundational unsolved problem in geometric graph theory. The conjecture was first studied by Hill in the 1950s, and the displayed quantity is attained by the standard conjectured optimal drawings; the equality is not known for general $n$.