Equality of susceptibility and component-size thresholds for size rules

Let $ell geqslant 2$ and let ${\mathcal R}$ be an $ell$-vertex size rule. Define $t_{\mathrm{b}}^{{\mathcal R}}$ as the supremum of the set of $t \geqslant 0$ for which the susceptibility $S(G_{tn}^{{\mathcal R}})$ is bounded in probability, and define $t_{\mathrm{c}}^{{\mathcal R}}$ as the supremum of the set of $t \geqslant 0$ for which $L_1(G_{tn}^{{\mathcal R}})/n$ converges to $0$ in probability as $n\to\infty$. Equality of thresholds conjecture. One has

$$t_{\mathrm{b}}^{{\mathcal R}}=t_{\mathrm{c}}^{{\mathcal R}}.$$

More precisely, for any $t>t_{\mathrm{b}}^{{\mathcal R}}$ and $\varepsilon>0$, there exist $\delta,n_0>0$, depending only on ${\mathcal R},\ell,t,\varepsilon$, such that

$$\mathbb{P}\bigl(L_1(G_{tn}^{{\mathcal R}})\geqslant\delta n\bigr)\geqslant1-\varepsilon$$

for all $n\geqslant n_0$. This is the finite-random-graph analogue of the equality of susceptibility and percolation thresholds in classical percolation theory. The source presents the assertion as a belief for size rules and gives no resolution, so its status remains open.