The setup of this question is very similar to a recent question I asked.
Let $p(z)$ be a polynomial of degree $n$ with complex zeros $\{z_i\}_{i=1}^n \subset\mathbb{C}$. Let $Z\subset \mathbb{C}$ be the set near the zeros, $$ Z = \bigcup_{i=1}^n D_{(100n)^{-1}}(z_i), $$ where $D_r(x)\subset \mathbb{C}$ is the disk of radius $r$ centered at $x$. Let $\gamma$ be the line segment $[-1,1]$ on the real axis in the complex plane.
I am interested in whether the inequality $$ \sup_{\gamma\cap Z} |p(x)| \leq \sup_{x\in \gamma\setminus Z} |p(x)| $$ holds. In other words, is it possible that the maximum value of $p$ on the line segment $\gamma$ takes place in $Z$?
It seems that the maximum principle might be useful here, but I am only interested in what happens to the polynomial on a line segment (which does not contain any disks in the complex plane). I am not sure how to proceed otherwise.
The answer is no! Consider the Chebyshev polynomial $T_n(x)$ with roots $$ x_k = \cos\left(\frac{2k-1}{2n}\pi\right), $$ and which satisfies $|T_n(x)| \leq T_n(1)$. The $n$-th root is a distance $\Theta(n^{-2})$ away from the maximum value.