Cut locus on a hypercube

Question

Inspired by the question, "Shortest path connecting two opposite points on a cube":

Q. What does the cut locus with respect to one corner of a hypercube in $\mathbb{R}^d$ look like?

"The cut locus from a point $p$ on a surface is the closure of the set of all points that have at least two shortest paths connecting them to $p$."¹

Here it is for $d=3$:

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          ^{Cut locus with respect to vertex $0$ is a star-tree rooted at vertex $6$.}

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¹Itoh, Jin-ichi, and Robert Sinclair. "Thaw: A tool for approximating cut loci on a triangulation of a surface." Experimental Mathematics 13.3 (2004): 309-325.