Dual Smale's mean value conjecture for complex polynomials

Let $P$ be a complex polynomial of degree $n\ge2$ satisfying $P(0)=0$ and $P'(0)=1$. A point $\zeta$ is a critical point of $P$ when $P'(\zeta)=0$. Dual Smale's mean value conjecture. There exists a critical point $\zeta$ of $P$ such that

$$\left|\frac{P(\zeta)}{\zeta}\right|\ge \frac1n.$$

This is the dual counterpart of Smale's mean value conjecture and was independently proposed by Dubinin--Sugawa and Ng. The paper proves it for odd polynomials with nonzero linear term, while the stated conjecture for general complex polynomials remains open.