Folklore conjecture on pseudorandom hypergraphs and the shattering threshold

A pseudorandom class of hypergraphs is a hereditary class whose sparse random model contains, with high probability, an induced subhypergraph on at least $(1-o_d(1))n$ vertices belonging to the class. Let the shattering threshold be the size threshold at which the solution space of independent sets in a sparse random $k$-uniform hypergraph fractures into exponentially many clusters. Folklore conjecture. There exists a pseudorandom class of hypergraphs whose independence number matches the shattering threshold. This is known for various pseudorandom graph classes, but had remained largely unexamined for higher uniformities; the paper states that uncrowded $k$-uniform hypergraphs resolve the question affirmatively for every $k\geq 3$.