I recently found out (Piranian, "The Shape of Level Curves") that a polynomial tract (ie a set of the form $G=\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) need not be starshaped with respect to the zeros of $p$ contained in $G$. This disappointed me bitterly, as that starshapeness was a pivotal step in a proposed "proof" I had of Smale's conjecture.

The places where $G$ is not starshaped with respect to the zeros of $p$ in $G$ are near critical points of $p$ in $G$, so I still hold out a tiny bit of hope for the starshapeness of $G$ with respect to the critical points of $p$ contained in $G$:

Conjecture: If $G$ is a tract of $p$ with smooth boundary, then $G$ is starshaped with respect to the critical points of $p$ contained in $G$.

Intuitions/proofs/disproofs/references are all welcome.