Shattering-threshold conjecture for independent sets in linear hypergraphs

Let $k\geq 3$ be an integer. A $k$-uniform hypergraph is linear if it contains no $2$-cycle, and let $\alpha({\mathcal H})$ denote its independence number. Shattering-threshold conjecture. For every $\varepsilon>0$, there exists $\Delta_0\in\mathbb{N}$ such that, for every $\Delta\geq\Delta_0$ and $n\in\mathbb{N}$, every $n$-vertex $k$-uniform linear hypergraph ${\mathcal H}$ of maximum degree $\Delta$ satisfies

$$\alpha({\mathcal H})\geq (1-\varepsilon)n\left(\frac{1}{k-1}\frac{\log\Delta}{\Delta}\right)^{\frac{1}{k-1}}.$$

This is presented as a stronger form of the linear-hypergraph conjecture above. Existing approaches rely on random sampling to reduce to the high-girth setting, and the source says that resolving it requires substantially new ideas.