Arjevani et al.'s lower-bound conjecture for polynomial iterations

Let $q(z)$ be a degree-$p$ monic real polynomial such that $q(1)=0$. Let $r(z)$ be any polynomial of degree $p-1$, and let $0<bc<\ell$. Then there exists $\nu\in[\mu,\ell]$ such that

$$\rho(q(z)-\nu\cdot r(z))\geq\frac{\sqrt{\ell/\mu}-1}{\sqrt{\ell/\mu}+1}.$$

Arjevani et al.'s lower-bound conjecture. For every such $q$, $r$, and $\mu$, some $\nu\in[\mu,\ell]$ satisfies the stated spectral-radius lower bound. This conjecture concerns tight lower bounds for stationary polynomial methods, and the surrounding paper proves it; the source does not provide further status evidence in the candidate metadata.