The weakly sparse large-treewidth subclass meta-conjecture

Let $\Pi$ be a graph-class property. A hereditary weakly sparse class is a hereditary graph class excluding some biclique $K_{t,t}$ as a subgraph. Meta-conjecture-ws$(\Pi)$. Every hereditary weakly sparse class $\mathcal C$ of unbounded treewidth contains a subclass $\mathcal C'\subseteq\mathcal C$ of unbounded treewidth with property $\Pi$. The paper refutes this for every finitely-hereditary or treewidth-hereditary property $\Pi$, while properties such as bounded twin-width are not refuted by the construction.