Smale's mean value conjecture

Let $P\in\mathbb{C}[\mathbf{z}]$ be a polynomial of degree $n+1\ge 2$ such that $P(0)=0$ and $P'(0)=1$, and let $b_1,\ldots,b_n$ be its critical points.

Smale's mean value conjecture.

$$\min\left\{\left|\frac{P(b_i)}{b_i}\right|:i=1,\ldots,n\right\}\le \frac{n}{n+1}.$$

This is equivalent to Smale's question of replacing the constant $4$ in his mean value theorem by $n/(n+1)$. It is verified for $n=1,2,3$, and the upper bound is attained by $P(z)=z^{n+1}+z$; the general case remains open.