Linear-size Borel independent complete sections conjecture

Let $H$ be a Borel graph with finite asymptotic separation index, and let $G_1$ and $G_2$ be spanning Borel subgraphs of $H$. A set is a complete section for a graph if it meets every connected component, and a section is $G_1$-independent if it contains no edge of $G_1$.

Linear-size Borel independent complete sections conjecture. There exist constants $C,0>0$ with the following property: if the maximum degree of $G_1$ is at most $0$ and every component of $G_2$ contains at least $C0$ vertices, then there is a Borel $G_1$-independent complete section for $G_2$.

The preceding theorem proves the analogous conclusion with a superlinear lower bound $2^{25}0\log0$ on the sizes of the components of $G_2$. The conjecture asks whether a linear bound in $0$ suffices; the finite asymptotic separation index assumption is necessary in general.