  Let $Q_n$ be the $n$-dimensional hypercube, and let ${\rm cr}(Q_n)$ be the
\textit{crossing number} of $Q_n$. Erd\H{o}s and Guy in 1973 conjectured the
following equality: ${\rm
cr}(Q_n)=\frac{5}{32}4^n-\lfloor\frac{n^2+1}{2}\rfloor 2^{n-2}$. In this paper,
we construct a drawing of $Q_n$ with less crossings when $n>6$, which implies
that for $n>6$ we have a strict inequality.
