$K_{5,5}$ is fully reconstructible in $\mathbb{C}^3$

A graph $G$ is fully reconstructible in $\mathbb{C}^d$ if the graph is determined from its $d$-dimensional measurement variety. The full reconstructibility problem has been solved for $d=1$ and $d=2$. For $d=3$, some necessary and some sufficient conditions are known and $K_{5,5}$ falls squarely within the gap in the theory. In this paper, we show that $K_{5,5}$ is fully reconstructible in $\mathbb{C}^3$.