The simplicial half-net conjecture for point sets

Let $S$ be a finite point set in $cmathbb{R}^d$. A simplicial half-net is a family of simplices determined by points of $S$ such that every halfspace containing at least half the points of $S$ contains one of these simplices. Simplicial half-net conjecture. Every point set $S\subset\mathbb{R}^d$ has a simplicial half-net. The conjecture would extend the lower-bound method based on simplicial half-nets from the planar setting to dimension $d$, potentially removing the restriction $k<\lfloor n/(d+1)\rfloor$ from the lower bound for the number of $(\leq k)$-facets.