Optimal colorful homogeneous selection with volume

Let $cmathcal F_1,\dots,\mathcal F_{d+1}$ be collections of volume-$1$ convex sets in $cmathbb R^d$, with $|\mathcal F_i|=n$ for every $i\in[d+1]$. A colorful homogeneous selection conjecture. If $n$ is sufficiently large, then there are subcollections $cmathcal{\widetilde F}_i\subseteq\mathcal F_i$ and a set $E$ with volume at least $v_d>0$ such that $|\widetilde{\mathcal F}_i|\geq c_d|\mathcal F_i|$ and, for every choice $cmathcal{\widetilde F}_i\in\widetilde{\mathcal F}_i$, one has

$$E\subseteq\operatorname{conv}(\widetilde F_1,\dots,\widetilde F_{d+1}).$$

This would reduce the currently used number of color classes to the optimal-looking value $d+1$; the paper notes that the necessity of the larger number of color classes is doubtful.