Global maximality conjecture for the function d on S(n,0)

Let $S(n,0)$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk and which vanish at $0$. For $p\in S_n$, define

$$d(p)=\max_{z\in Z(p)}\min_{w\in Z(p')}|z-w|.$$

The paper has determined a class of $0$-maximal polynomials of degree $n$. Global maximality conjecture for $S(n,0)$. If $p\in S(n,0)$, then

$$d(p)\le \left(\frac{1}{n}\right)^{\frac{1}{n-1}},$$

and equality occurs if and only if $p$ is one of the polynomials in the stated classification theorem. This would show that the $0$-maximal polynomials are global maxima of $d$ on $S(n,0)$, strengthening the local result proved in the paper. The inequality and its equality classification are left as a conjecture.