Weak Borsuk conjecture for polytopes

Let $P\subseteq\mathbb R^d$ be a polytope with $m$ facets. A covering by smaller-diameter sets is a cover of $P$ by sets whose diameters are strictly smaller than the diameter of $P$. Weak Borsuk conjecture for polytopes. Every polytope $P$ with $m$ facets can be covered by $m$ sets of smaller diameter. The source says that a positive answer would give an alternative route concerning projections of the cut polytope, but gives no resolution status.