Asymptotic corank growth conjecture for chromatic roots

For a finite graph $G$, let $\pi(G,z)$ be its chromatic polynomial and define

$$\rho(G)=\max\{|z-1|:\pi(G,z)=0\}.$$

Let $\rho_k$ be the maximum of $\rho(G)$ over graphs of corank $k$.

Asymptotic corank-growth conjecture. As $k\to\infty$,

$$\rho_k=\left[1+o(1)\right]\frac{k}{\log k}.$$

This is presented as an expected consequence of the corrected fixed-corank extremal conjecture. The paper establishes matching lower bounds of this order from generalized theta graphs and an upper bound $\rho_k\leq k$, but does not determine the asymptotic growth exactly.