On a metric property of analytic sets

Let $H$ be an algebraic set in $\mathbf C^n$ containing the origin and let $S=\{z\in\mathbf C^n:|z|=1\}$ be the unit sphere.
Conjecture. The diameter of one of the connected components of $H\cap S$ is greater than one.
In this article it is shown that this is false if the requirement that $H$ be algebraic is weakened to the demand that the projections onto the coordinate planes be open. If, however, $S$ is replaced by the boundary of the unit polydisc, then the conjecture holds and the proof uses only the openness of the projection.
Bibliography: 3 titles.