Strong cross-star conjecture for hereditary families

Let $1\leq t\leq r$, let $\emptyset\neq S\subseteq [r]$, and let $\mathcal{H}$ be a hereditary family with covering number $\mu(\mathcal{H})\geq (t+1)(r-t+1)$. Write $\mathcal{H}^{(S)}$ for the subfamily consisting of members whose sizes lie in $S$, and use the strong cross-$t$-star property as in the paper. Hereditary-family cross-star conjecture. The pair

$$(\mathcal{H}^{(S)},\mathcal{H}^{(S)})$$

has the strong cross-$t$-star property. The paper additionally conjectures that the pair has the extrastrong cross-$t$-star property when $\mu(\mathcal{H})>(t+1)(r-t+1)$.