Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter?
Background and motivation
The Borsuk conjecture (disproved in 1993) asserted that every set of diameter 1 in $R^d$ can be covered by $d+1$ sets of smaller diameter. Since every $d$-polyope has at least $d+1$ facets our proposal is indeed a weaker statement.
My motivation for this question comes from the paper Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds by Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary and Ronald de Wolf. The paper solves an old conjecture of Yannakakis about projections of polytopes and shows, in particular, that the $n$-dimensional cut polytope cannot be described as a projection of a polytope with only a polynomial number (in $n$) of facets.
Another proof of this result of Fioroni et als. (regarding cuts polytopes) will follow from an affirmative answer of a slight strengthening of the proposed problem:
Problem: If $P$ is a $d$-polytope of diameter 1 with $n$ facets then $P$ can be covered by $n$ sets of diameter $1-\epsilon$ wher $\epsilon$ may depend on $n$ (but not on $P$.)