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[Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove that there exist $\\Omega \\left(2^d\\right)$ crossing pairs of hyperedges in such a drawing of $K_{2d}^d$. We improve this lower bound by showing that there exist $\\Omega \\left(2^d \\sqrt{ d}\\right)$ crossing pairs of hyperedges in a $d$-dimensional rectilinear drawing of $K_{2d}^d$. We also prove the following results.\n  1. 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