TITLE: Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial
QUESTION [10 upvotes]: The Gauss--Lucas Theorem states that all zeros of derivative of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$.  By iteration, this implies that the zeros of $p',p^{(2)},\ldots,p^{(n-1)}$ are contained in the convex hull of the zeros of $p$.
The Riemann--Hurwitz Theorem (among others) implies that if a tract $D$ of $p$ (namely a component of the set $\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) contains all the zeros of $p$ in its bounded face, then all the critical points of $p$ are contained in $D$.
My conjecture is that in fact, if $D$ is a tract of $p$ and contains all the zeros of $p$, then $D$ also contains all the zeros of $p',p^{(2)},\ldots,p^{(n-1)}$.
This certainly does not follow by straight-forward iteration, since in general there may not exist some tract $D'$ of $p'$ which contains all the zeros of $p'$ in $D$, and is itself contained in $D$.  It seems that the tracts and level curves of $p'$ do not interact very nicely with the tracts and level curves of $p$ (even worse for $p'',p''',\ldots$).
This question was originally asked on M.SE, where it received no answers, however user Behavior did point out that this conjecture could be equivalently stated as:
Conjecture: Let $M=\max(|p(z)|:p'(z)=0)$, then if $w$ is a zero of any derivative of $p$, then $|p(w)|\leq M$.

REPLY [7 votes]: Very interesting problem. At first I thought the statement might be true for real valued polynomials, but unfortunately I found the following, 

Unless I am misunderstanding the problem, I believe this is the counterexample you were looking for. Although, we note that the second derivative does look suspiciously still trapped inside of the tract. Maybe, your conjecture holds for only the first and second derivative.