Conjecture on an n-independent convergence rate for Kalantari's bound

Let $p(z)$ be a polynomial of degree $n$, let $\rho_{\min}:=\min_{\rho:p(\rho)=0}|\rho|$, and let $L_m$ be the lower bound in the inequality $L_m/\rho_{\min}\geq 1-\epsilon$. n-independent convergence-rate conjecture. There is a fixed constant $c$ with

$$1<c\ll 2$$

such that the inequality holds for $m=m_c$, where

$$m_c=O\left(\frac{1}{\epsilon^c}\right),$$

and $m_c$ does not depend on $n$. This conjecture refines the experimentally observed rate by asserting independence from the polynomial degree; the paper does not provide a proof, and the exponent and constant remain to be established.