Strong modeling limit conjecture for monadically stable graph classes

Let $\mathscr C$ be a monadically stable class of graphs. An FO-convergent sequence is a sequence whose first-order formula densities converge, and a strong modeling is a modeling satisfying the strong finitary mass transport principle: for all measurable subsets $A$ and $B$, if every vertex of $A$ has at least $a$ neighbors in $B$ and every vertex of $B$ has at most $b$ neighbors in $A$, then

$$a\,\nu(A)\leq b\,\nu(B).$$

Strong modeling limit conjecture. Every FO-convergent sequence of graphs in $\mathscr C$ has a strong modeling FO-limit. The paper proves existence of modeling FO-limits for monadically stable classes, while the stronger mass-transport conclusion is known only for restricted classes of graphs according to the cited literature.