Weak convergence conjecture for rescaled chromatic measures of dense graph sequences

Let $(G_n)_n$ be a sequence of graphs convergent in the dense model. For a graph $G$ of order $n$, let $\nu_G$ denote its rescaled chromatic measure, defined from the chromatic-root measure $\mu_G$ by $\nu_G(X)=\mu_G(nX)$ for $X\subset\mathbb{C}$. Weak convergence conjecture. The sequence of probability measures $\nu_{G_n}$ converges weakly. This would extend the convergence of holomorphic moments to weak convergence and would allow chromatic roots to be associated with graphons; beyond the stated moment convergence, the paper reports only the absence of counterexamples as support. It includes, as a weak special case, convergence for Erdős–Rényi random graphs with constant edge probability and for random graphs sampled from a graphon.