The polylogarithmic susceptibility conjecture for bounded-degree graphs

Fix a maximum degree $d$. Let $(G_n)_{n\geq1}$ be a sequence of finite connected graphs, with $G_n=(V_n,E_n)$, $|V_n|\to\infty$, and maximum degree at most $d$. The polylogarithmic susceptibility conjecture. There exist constants $C_{d,\lambda}>0$ and $\ell>0$ such that

$$\lim_{n\to\infty}\mathbb{P}_{\lambda}\left[\mathcal{S}(G_n)\leq C_{d,\lambda}\log^{\ell}|V_n|\right]=1.$$

The conjecture asserts a uniform polylogarithmic upper bound on susceptibility for bounded-degree graph sequences. The source further speculates on particular values of $\ell$ and the dependence of the constant, but those stronger speculations are not included in this row because they are presented as suspicions rather than the stated conjecture.