Maximum colourful simplicial depth conjecture

Let a colourful configuration consist of $(d+1)$ points in each of $(d+1)$ colours in $d$-dimensional space. Its core is the intersection of the convex hulls of the individual colours. For a point $p$ in the interior of the core, the colourful simplicial depth is the number of colourful simplices containing $p$, where a colourful simplex has one vertex of each colour. Let $u(d)$ denote the maximum of this depth over such configurations and points.

Maximum colourful simplicial depth conjecture. The maximum colourful simplicial depth of any point in the interior of the core is

$$\nu(d)=d^{d+1}+1.$$

A construction gives the lower bound $\nu(d)\geq d^{d+1}+1$, and the conjecture asserts that this bound is tight; the authors report that a computer search found no counterexamples.