Colorful quantitative volume Helly conjecture

Let $\mathcal{F}_1, \dots, \mathcal{F}_{2d}$ be finite families of convex sets in $\mathbb{R}^d$. Assume that for any choice $F_1 \in \mathcal{F}_1, \dots, F_{2d} \in \mathcal{F}_{2d}$, the volume of the intersection

$$\bigcap\limits_{i \in [2d]} F_i$$

is at least $1$. Colorful quantitative volume Helly conjecture. Then for some $i \in [2d]$, the intersection of all sets in the family $\mathcal{F}_i$ has volume at least $C_d$, for some strictly positive constant $C_d$ depending only on the dimension $d$. This is the open volumetric analogue of the colorful quantitative diameter Helly theorem; the expected lower bound is asserted only to depend positively on the dimension.