Quantitative colorful Tverberg conjecture with $2d$ color classes

For integers $d,r\geq 1$, let $cmathcal F_1,\dots,\mathcal F_{2d}$ be families in $cmathbb R^d$, each containing $f(d,r)$ convex sets of volume $1$. A colorful Tverberg conjecture. For any such $2d$ families, there are $r$ disjoint transversals $cmathcal T_1,\dots,\mathcal T_r$ such that

$$\operatorname{vol}\left(\bigcap_{i=1}^r \operatorname{conv}(\mathcal T_i)\right)\geq v_d>0.$$

A result of this kind would support quantitative colorful homogeneous selection, while the number of color classes and the required function $f(d,r)$ remain open.