Theta-graph extremal conjecture for chromatic roots at fixed corank

Let $G$ be a finite graph of corank $k$, and let

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

For $k\geq 1$, write $\rho_k$ for the maximum of $\rho(G)$ over graphs of corank $k$, and let $\rho^{(2,k+1)}$ denote the corresponding value for the $(k+1)$-ary generalized theta graph $\Theta^{(2,k+1)}$.

Fixed-corank theta extremal conjecture. If $G\neq K_4$, then

$$\rho(G)\leq \rho^{(2,k+1)}.$$

In particular, for $k\geq 4$,

$$\rho_k=\rho^{(2,k+1)}.$$

The originally proposed statement without the exception $G=K_4$ is refuted: $K_4$ has corank $3$ and a chromatic root at $z=3$, whereas $\rho^{(2,4)}\approx 1.9636$. The corrected conjecture remains open in the supplied source.