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

Let $X$ be a $GC_{d,n}$ set in ${\mathbb R}^d$: it has ${n+d\choose d}$ points, and for every $p\in X$ there is a product of $n$ linear forms that equals $1$ at $p$ and $0$ at every other point of $X$. A hyperplane is maximal if it contains ${d-1+n\choose n}$ points of $X$. de Boor's conjecture. Every $GC_{d,n}$ set contains a maximal hyperplane. This is one of de Boor's proposed higher-dimensional generalizations of the Gasca–Maeztu conjecture; the source gives no resolution.