The transversal-dimension conjecture for colorful Helly families

Let $d$ be a positive integer, and let $\mathcal{F}=\mathcal{F}_1\cup\cdots\cup\mathcal{F}_d$ be a $d$-colored family of convex sets in $\mathbb{R}^d$. Write ${{\EuScript{CH}}}(\mathcal{F}_1,\dots,\mathcal{F}_d)$ for the Colorful Helly hypothesis. A family is crossed by $r$ $k$-flats if every member of the family meets at least one of those $r$ flats.

Transversal-dimension conjecture. For every integer $k$ with $1\leq k\leq d$, there exist numbers $h(k,d)$ such that, for every $d$-colored family $\mathcal{F}$ satisfying ${{\EuScript{CH}}}(\mathcal{F}_1,\dots,\mathcal{F}_d)$, there exist numbers $k_1,\dots,k_d$ satisfying

$$\sum_{1\leq i\leq d} k_i\leq d$$

and each color class $\mathcal{F}_i$, for $i\in[d]$, can be crossed by $h(k_i,d)$ $k_i$-flats.

This conjecture proposes a stronger structural consequence of the Colorful Helly hypothesis than the results established earlier in the paper, by simultaneously controlling the transversal dimensions of all color classes. The supplied text does not state whether the conjecture is open or resolved.