Kim's refined colorful fractional Helly conjecture

Let $d$ be a non-negative integer. For each $i\in[d+1]$, let $n_i$ be positive and let $r_i$ be a non-negative integer with $n_i\ge r_i+1$. Let $\mathcal{F}_1,\dots,\mathcal{F}_{d+1}$ be families of convex sets in $\mathbb{R}^d$ such that $|\mathcal{F}_i|=n_i$ and no subfamily of $\mathcal{F}_i$ of size $r_i+1$ has nonempty intersection. Kim's refined conjecture. The number of colorful $(d+1)$-tuples with nonempty intersection is at most

$$n_1\cdots n_{d+1}-(n_1-r_1)\cdots(n_{d+1}-r_{d+1}).$$

This is a refinement proposed as a possible route to the optimal colorful fractional Helly theorem; the source presents it as Kim's conjecture, and its resolution is not supplied here.