The fractional Helly number conjecture for families in Euclidean space

Let $b\geq 1$, $d\geq 2$ be integers, and let $\alpha\in(0,1)$. For a family $\mathcal{A}$ of sets in $\mathbb{R}^d$, write $TC_{\lceil d/2\rceil}(\mathcal{A})$ for its topological complexity in dimension $\lceil d/2\rceil$. The fractional Helly number conjecture. For any integers $b\geq 1$, $d\geq 2$ and $\alpha\in(0,1)$, there exists $\beta=\beta(\alpha,b,d)>0$ such that, whenever $\mathcal{A}$ is a family of $n\geq d+1$ sets in $\mathbb{R}^d$ with $TC_{\lceil d/2\rceil}(\mathcal{A})\leq b$ and at least $\alpha\binom{n}{d+1}$ of its $(d+1)$-tuples intersect, $\mathcal{A}$ contains an intersecting subfamily of size at least $\beta n$. The conjecture predicts that the fractional Helly number for families in $\mathbb{R}^d$ is $d+1$, extending the optimal surface case discussed immediately before it; its status is not resolved in the supplied text.