The biplanar crossing number conjecture for the 8-dimensional hypercube

Let $Q_8$ be the 8-dimensional hypercube, and let $cr_2(Q_8)$ denote its biplanar crossing number, the minimum total number of crossings in a drawing of $Q_8$ on two planes. Biplanar crossing number conjecture.

$$cr_2(Q_8)=128.$$

The paper proves the upper bound $cr_2(Q_8)\leq 128$ by constructing a two-plane drawing with at most $64$ crossings in each plane. The conjecture asserts that this bound is sharp; the authors state that a proof of the matching lower bound remains elusive.