Holroyd–Talbot conjecture for levels of hereditary families

Let $\mathcal{H}$ be a hereditary family, let $\mu(\mathcal{H})$ denote the size of a smallest base of $\mathcal{H}$, and let $\mathcal{H}^{(r)}$ be its $r$th level. The $r$th level has the $t$-star property when its largest $t$-intersecting subfamilies are $t$-stars. Holroyd–Talbot conjecture. If $1 \leq t \leq r$ and

$$\mu(\mathcal{H}) \geq (t+1)(r-t+1),$$

then $\mathcal{H}^{(r)}$ has the $t$-star property. This generalizes the Holroyd–Talbot conjecture and includes the case of the power-set levels; it remains open in the stated generality.