Jung–Keszegh–Pálvölgyi–Yuditsky conjecture on piercing maximum hypergraph cliques

Let $[n]=\{1,\ldots,n\}$, let $\binom{[n]}{k}$ denote the family of $k$-element subsets of $[n]$, and let $\mathcal{H}\subset\binom{[n]}{k}$ be a $k$-uniform hypergraph. Write $v(\mathcal{H})$ for its number of vertices and $\omega(\mathcal{H})$ for the size of its largest clique.

Jung–Keszegh–Pálvölgyi–Yuditsky conjecture. For all $k\geq2$ there exists a constant $c_k<1$ such that, whenever

$$\omega(\mathcal{H})>c_kv(\mathcal{H}),$$

the maximum-size cliques of $\mathcal{H}$ can be hit with a single point.

The conjecture proposed a hypergraph analogue of Hajnal's graph theorem, asserting that sufficiently large maximum cliques force a common piercing vertex. It is refuted in the source: for every $c<1$, $k\geq3$, and $t\geq1$, there are $k$-uniform hypergraphs with maximum clique size exceeding $c$ times the number of vertices whose maximum cliques cannot be pierced by $t$ vertices.