Shadow-boundary conjecture for strongly generic polynomials

Let $P$ be a strongly generic polynomial of degree $d e$ at least $3$, with roots in convex position but not forming a regular polygon. Let $\x1d4fe_P$ be the shadow of $P$, and let $\x0043\x006f\x006e\x0076_P$ be the convex hull of its roots. For $\x03b1\x2208[0,d]$, define

$$F_\x03b1(z)=z-\x03b1\frac{P(z)}{P'(z)}.$$

Shadow-boundary conjecture. (i) $\x1d4fe_P\subset\x0043\x006f\x006e\x0076_P$ is a concave domain. (ii) The boundary of $\x1d4fe_P$ is contained in the union, over $\x03b1\x2208[0,d]$, of the critical values of $F_\x03b1$ with respect to $z$; equivalently, it consists of the values of $u$ for which $\x03a6(\x03b1,z,u)=\x03b1P(z)+(u-z)P'(z)$ has a multiple root in $z$ for some fixed $\x03b1$. The claim describes the geometry of the support domain arising from asymptotic derivative root measures; the source gives no proof or resolution.