Shapiro's shadow-domain conjecture for derivatives of polynomial powers

Let $P$ be a polynomial of degree $d\geq 3$ whose roots are in convex position and do not form a regular polygon. For $0\leq m\leq nd$, set

$$Q_{n,m}(z)=\frac{d^mP^n(z)}{dz^m},$$

and let $\Upsilon_P$ be the closure of the union of all roots of these polynomials. For $\boldsymbol{\alpha}\in[0,d]$, define

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

Shapiro's shadow conjecture. The set $\Upsilon_P$ is a closed domain in the convex hull of the roots of $P$; all critical points of $P$ lie on its boundary; its boundary has no inflection points; and its boundary is contained in the union of all critical values, with respect to $z$, of $F_{\boldsymbol{\alpha}}$ as $\boldsymbol{\alpha}$ ranges over $[0,d]$. The conjecture describes the geometric shadow formed by roots of derivatives of powers of $P$; the supplied text gives no resolution.