Aldous–Lyons-type characterization of finite-graph limits

Let $\mathbf L$ be a modeling. Assume: (i) the theory of $\mathbf L$ has the finite model property; (ii) every interpretation of $\mathbf L$ satisfies the finitary mass transport principle, namely, for first-order formulas $\alpha,\beta,\gamma$ satisfying

$$\alpha(x)\vdash (\exists y_1\dots y_b)\,\bigwedge_{i=1}^b \Bigl(\gamma(y_i,x)\wedge\beta(y_i)\wedge\bigwedge_{i<j\leq b}(y_i\neq y_j)\Bigr)$$ $$\beta(x)\vdash \neg(\exists y_1\dots y_{a+1})\,\bigwedge_{i=1}^{a+1} \Bigl(\gamma(x,y_i)\wedge\alpha(y_i)\wedge\bigwedge_{i<j\leq a+1}(y_i\neq y_j)\Bigr),$$

one has

$$b\,\langle\alpha,\mathbf L\rangle\leq a\,\langle\beta,\mathbf L\rangle;$$

and (iii) for every integer $d$ there is an integer $N$ such that $\mathbf L$ does not contain the $d$-th subdivision of $K_N$. Aldous–Lyons-type conjecture. Under these assumptions, $\mathbf L$ is the FO-limit of a sequence of finite graphs. This is proposed as a generalization of the Aldous–Lyons conjecture; the supplied text does not state that it has been proved or refuted.