Fractional-to-integral transversal bound for fat convex sets

Let $4\mathcal{H}4$ be a family of families of $k$-flats, where each member of $4\mathcal{H}4$ consists of $k$-flats intersecting members of a family of $4\rho4$-fat convex sets in $4\mathbb{R}^d4$. Let $4\tau(\mathcal{H})4$ denote its transversal number and $4\tau^*(\mathcal{H})4$ its fractional transversal number.

Fractional transversal conjecture. There is a function $f$ such that

$$\tau(\mathcal{H}) \leq f\bigl(\tau^*(\mathcal{H})\bigr).$$

Such a bound would provide the missing fractional-to-integral step for $k$-transversals of $4\rho4$-fat convex sets. The analogous result is known for bounded VC-dimension families and for suitable intersection-closed families, but the paper states that neither available theorem applies in this setting; the source gives no resolution.