Conjecture on local maxima for Sendov's conjecture

Let $S_n$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk, and let

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

A polynomial is locally maximal for Sendov's conjecture if it is a local maximum of $d$ on $S_n$. The local-maxima conjecture. The polynomials $z^n+e^{i\theta}$, with $\theta\in\mathbf{R}$, are all the local maxima for $d$; equivalently, a polynomial $p\in S_n$ is locally maximal for Sendov's conjecture if and only if $p(z)=z^n+e^{i\theta}$ for some $\theta\in\mathbf{R}$. The source presents this as a stronger version of the extremal-polynomial conjecture and gives no resolution.