Colorful Bárány–Katchalski–Pach diameter conjecture

Let $\mathcal{F}_1,\ldots,\mathcal{F}_{2d}$ be finite families of convex sets in $\mathds{R}^d$. If $\bigcap_{i=1}^{2d}F_i$ has diameter greater than or equal to $1$ for every $2d$-tuple $(F_i)_{i=1}^{2d}$ with $F_i\in\mathcal{F}_i$, then there exists an index $k\in[2d]$ such that $\bigcap\mathcal{F}_k$ has diameter greater than or equal to $c d^{-1/2}$, for some absolute constant $c$.

Colorful Bárány–Katchalski–Pach conjecture. Under these hypotheses, there is an index $k\in[2d]$ whose entire family has intersection of diameter at least $c d^{-1/2}$, for some absolute constant $c$.

This is the colorful analogue of the Bárány–Katchalski–Pach diameter conjecture, with each family viewed as a color class. The source postulates this extension after proving a related colorful fractional theorem; its resolution status is not specified in the supplied text.