Lovász–Szegedy Borel representation conjecture for Lebesgue graphs

Let $\mathbf L$ be a Lebesgue graph, meaning a graph on the standard probability space $[0,1]$ whose edge set is Lebesgue-measurable. A finite graph $F$ has a density in a graph given by the measure of the corresponding induced-copy event. Lovász–Szegedy conjecture. For every Lebesgue graph $\mathbf L$ there exists a Borel graph $\mathbf B$ such that, for every finite graph $F$, the density of $F$ in $\mathbf L$ and $\mathbf B$ are equal, and every finite induced subgraph of $\mathbf B$ has positive density in $\mathbf B$. This is a Borel regularization conjecture for graphons or measurable graph structures; the paper proves a related removal lemma for monadically stable structures, but does not establish this general graph statement.