Slicing all Edges of an $n$-cube Requires $n^{2\mathrm{/}3}$ Hyperplanes

Consider the $n$-cube graph with vertices $\{-1,1{\}}^n$ and edges connecting vertices with hamming distance $1$. How many hyperplanes in ${\mathbb{R}}^n$ are needed in order to dissect all edges? We show that at least $\tilde{\mathrm{\Omega }}(n^{2\mathrm{/}3})$ are needed, which improves the previous bound of $\mathrm{\Omega }(n^{0.51})$ by Yehuda and Yehudayoff.