Strong cross-star conjecture for lifted set families

Let $t\geq 1$, let $\mathcal{F}$ be a family, and let $m$ be a positive integer. The notation $\mathcal{S}_{\mathcal{F},m}$ denotes the associated lifted set family, and the strong cross-$t$-star property is the property asserted in the conjecture. Strong cross-star conjecture. There exists a positive integer $m_0(t)$ such that, for every family $\mathcal{F}$ and every $m\geq m_0(t)$, the pair

$$(\mathcal{S}_{\mathcal{F},m},\mathcal{S}_{\mathcal{F},m})$$

has the strong cross-$t$-star property. The paper further conjectures that the smallest possible value is $m_0(t)=t+1$, and that the pair has the extrastrong cross-$t$-star property when $m>t+1$.