The colourful simplicial depth conjecture

Given $d+1$ sets of points, or colours, $S_1,\dots ,S_{d+1}$ in $\mathbb R^d$,acolourfulsimplexisaset$T\subseteq\bigcup_{i=1}^{d+1}S_i$suchthat$|T\cap S_i|\leq 1$,forall$i\in{1,\ldots,d+1}$. The colourful Carath'eodory theorem states that, if $\mathbf{0}$ is in the convex hull of each $S_i$, then there exists a colourful simplex $T$ containing $\mathbf{0}$ in its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597–604 (2006)) conjectured that, when $\mathrm{\mid }{S}_i\mathrm{\mid }=d+1$ for all $i\in \{1,\dots ,d+1\}$, there are always at least $d^2+1$ colourful simplices containing $\mathbf{0}$ in their convex hulls. We prove this conjecture via a combinatorial approach.