Codenotti–Santos–Schymura conjecture on covering radii of non-hollow lattice polytopes

Let $P\subseteq \mathbb{R}^d$ be a non-hollow lattice polytope. A full-dimensional lattice polytope is called terminal if it is a direct sum of translates of terminal simplices, where the terminal simplex is $T_d=\operatorname{conv}(-\mathbb{1}_d,e_1,\dots,e_d)$. Codenotti–Santos–Schymura conjecture. The covering radius satisfies

$$\mu(P)\leqslant \frac{d}{2},$$

with equality if and only if $P$ is a terminal $d$-polytope up to a unimodular transformation. Codenotti, Santos and Schymura pose this as the problem of determining the maximal covering radius of a non-hollow lattice polytope and prove the conjecture in dimensions at most $3$; the general case remains open.