Holroyd–Talbot–Borg conjecture for intersecting faces of simplicial complexes

Let $\Delta$ be a simplicial complex whose smallest facet has $d$ vertices. For a vertex $v$ of $\Delta$, let $\operatorname{link}_{\Delta}v$ denote its link, and let $f_{r-1}(\operatorname{link}_{\Delta}v)$ be the number of faces of dimension $r-1$ in that link. Let $\mathcal{A}$ be a family of pairwise-intersecting faces of $\Delta$, each with $r$ elements.

Holroyd–Talbot–Borg conjecture. If $r\leq d/2$, then there is some vertex $v$ of $\Delta$ such that

$$|\mathcal{A}|\leq f_{r-1}(\operatorname{link}_{\Delta}v).$$

If equality holds and $r<d/2$, then $\mathcal{A}$ consists of the $r$-element faces containing some vertex $v$.

This extends Erdős–Ko–Rado-type results from uniform set systems and independence complexes of graphs to arbitrary simplicial complexes. The source presents it as the question raised by Holroyd and Talbot and extended by Borg; its resolution is not specified here.