I came accross a paper Faria, Herrera de Figueiredo, Sykora, Vrto: An improved upper bound on the crossing number of the hypercube that proves half of this, getting the correct upper bound.
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Improved upper bound
I came accross a paper Faria, Herrera de Figueiredo, Sykora, Vrto: An improved upper bound on the crossing number of the hypercube that proves half of this, getting the correct upper bound.