Miller's conjecture on maximal polynomials

Let $\beta\in\bar{D}$ and let $S(n,\beta)$ be the set of monic complex polynomials in $S_n$ having at least one zero at $\beta$. For $\alpha\in\mathbf{C}$ and $p\in S(n,\beta)$, define

$$|p|_\alpha=\min_{w\in Z(p')}|\alpha-w|.$$

The $\alpha$-critical circle is the circle centered at $\alpha$ with radius $|p|_\alpha$. Call $p$ maximal with respect to $\alpha$ in $S(n,\beta)$ if $|p|_\alpha\ge |q|_\alpha$ for every $q\in S(n,\beta)$. Miller's conjecture. If $p\in S(n,\beta)$ is maximal with respect to $\alpha$, then all zeros of $p'$ lie on the $\alpha$-critical circle and $p$ has as many zeros as possible on the unit circle. The conjecture concerns the structure of maximizers in the variational problem associated with Sendov's conjecture. The paper cites prior work on partial results; the general claim is left unresolved.