Polynomial spectral-radius lower-bound conjecture for p-CLI methods

Let $0<\mu<L$, and let $a(z)$ and $b(z)$ be polynomials of degree at most $p-1$ satisfying $b(1)=1$. For a polynomial, let $\rho$ denote the maximum modulus of its roots. Write

$$Q=\frac{L}{\mu}.$$

Polynomial lower-bound conjecture. There exists $\eta\in[\mu,L]$ such that

$$\rho\bigl(z^p-(\eta a(z)+b(z))\bigr)\geq\frac{\sqrt{L/\mu}-1}{\sqrt{L/\mu}+1}.$$

This is presented as a reduction of the algorithmic conjecture to a question about polynomials, so it expresses the same proposed obstruction in purely analytic terms. The source gives no evidence that it has been resolved.