If $p(z)$ is a polynomial of degree $n$ such that $\{z:|p(z)|\le 1\}$ is connected then is it true that$$\underset{\begin{array}{c}z\in \mathbb{C}\\ |p(z)|\le 1\end{array}}{\max }|p^{\prime }(z)|\le (\frac{1}{2}+o(1))n^2?$$