Second selection lemma over valued fields

For each $d\in\mathbb{N}_{\geq 1}$, let $c,s>0$ be constants. Second selection lemma over valued fields. For every valued field $K$, every $\alpha\in(0,1]$, every $n\in\mathbb{N}$, every $X\subseteq K^d$ with $|X|=n$, and every family $\mathcal{F}$ of $(d+1)$-element subsets of $X$ satisfying

$$|\mathcal{F}|\geq\alpha{n\choose d+1},$$

a point lies in the convex hulls of at least

$$c\alpha^s{n\choose d+1}$$

of the members of $\mathcal{F}$. The source presents this as a consequence that would formally follow from the colorful Tverberg conjecture, and does not indicate a resolution.