Brent's root-location conjecture for the interpolation polynomials

Let $a_n(x)$ be the irreducible factor occurring in the conjectured factorization of $A_n(x)$. Brent's root-location conjecture. If $n>1$, $a_n(\rho)=0$, and $\rho\neq\pm2$, then

$$|\rho|\leq \frac{n}{\log n}.$$

Consequently, even if the preceding nonvanishing conjecture is false, $\alpha_n(m)\neq0$ whenever $n\geq0$ and $m>\max(n/\log n,2)$. The conjecture also predicts that the roots of $a_n$ lie on a closed curve $P_n$ symmetric about both coordinate axes, or on an axis, with exactly two nonzero imaginary roots and exactly one self-intersection of $P_n$, at zero; numerical experiments are offered as evidence.