de Boor's multiple-maximal-hyperplane conjecture for geometrically characterized sets

Let $X$ be a $GC_{d,n}$ set in ${\mathbb R}^d$, and call a hyperplane maximal when it contains ${d-1+n\choose n}$ points of $X$. de Boor's multiple-maximal-hyperplane conjecture. Every $GC_{d,n}$ set contains at least $d+1$ maximal hyperplanes. This is the stronger of de Boor's two higher-dimensional generalizations described in the source; the source gives no resolution.